From Coq Require Import ssreflect ssrfun ssrbool.

From Ltac2 Require Import Ltac2.

From Coq Require Import Ensembles Bool String.

From Coq.Logic Require Import FunctionalExtensionality Eqdep_dec.
From Equations Require Import Equations.

Require Import Coq.Program.Tactics.

From MatchingLogic Require Import Syntax DerivedOperators_Syntax ProofSystem IndexManipulation wftactics.

From stdpp Require Import list tactics fin_sets coGset gmap sets.
New coercion path [Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]

From MatchingLogic.Utils Require Import stdpp_ext.
Import extralibrary.
From MatchingLogic Require Import Logic
  ProofInfo
  ProofMode_base 
  ProofMode_propositional
  ProofMode_firstorder
  ProofMode_fixpoint
  ProofMode_reshaper
.

Import MatchingLogic.Logic.Notations.

Set Default Proof Mode "Classic".

Open Scope ml_scope.
Open Scope string_scope.
Open Scope list_scope.

Ltac2 _callCompletedTransformedAndCast
  (t : constr) (transform : constr) (tac : constr -> unit) :=
  let tac' := (fun (t' : constr) =>
    let tac'' := (fun (t'' : constr) =>
      let tcast := open_constr:(@useGenericReasoning'' _ _ _ _ _ $t'') in
      fillWithUnderscoresAndCall tac tcast []
    ) in
    fillWithUnderscoresAndCall (fun t''' => tac'' t''') transform [t']
  ) in
  fillWithUnderscoresAndCall tac' t []
.

Ltac2 mlApplyMetaGeneralized (t : constr) :=
  _callCompletedTransformedAndCast t constr:(@reshape_lhs_imp_to_and_forward) _mlApplyMetaRaw ;
  try_solve_pile_basic ();
  try_wfa ()
.

Ltac _mlApplyMetaGeneralized t :=
  let ff := ltac2:(t' |- mlApplyMetaGeneralized (Option.get (Ltac1.to_constr(t')))) in
  ff t;
  rewrite [foldr patt_and _ _]/=
.


Tactic Notation "mlApplyMetaGeneralized" constr(t) :=
  _mlApplyMetaGeneralized t
.

#[local]
Example ex_mlApplyMetaGeneralized  {Σ : Signature} Γ a b c d e f:
  well_formed a ->
  well_formed b ->
  well_formed c ->
  well_formed d ->
  well_formed e ->
  well_formed f ->
  Γ ⊢ a ---> b ---> c ---> d ---> e ---> f ->
  Γ ⊢ (a and b and c and d and e) ---> f.
Σ: Signature
Γ: Theory
a, b, c, d, e, f: Pattern
well_formed a
→ well_formed b
  → well_formed c
    → well_formed d
      → well_formed e
        → well_formed f
          → Γ ⊢ a ---> b ---> c ---> d ---> e ---> f
            → Γ ⊢ a and b and c and d and e ---> f
Proof.
Σ: Signature
Γ: Theory
a, b, c, d, e, f: Pattern
well_formed a
→ well_formed b
  → well_formed c
    → well_formed d
      → well_formed e
        → well_formed f
          → Γ ⊢ a ---> b ---> c ---> d ---> e ---> f
            → Γ ⊢ a and b and c and d and e ---> f
  intros wfa wfb wfc wfd wfe wff H.
Σ: Signature
Γ: Theory
a, b, c, d, e, f: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
wfd: well_formed d
wfe: well_formed e
wff: well_formed f
H: Γ ⊢ a ---> b ---> c ---> d ---> e ---> f
Γ ⊢ a and b and c and d and e ---> f
  toMLGoal.
Σ: Signature
Γ: Theory
a, b, c, d, e, f: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
wfd: well_formed d
wfe: well_formed e
wff: well_formed f
H: Γ ⊢ a ---> b ---> c ---> d ---> e ---> f
well_formed (a and b and c and d and e ---> f)
Σ: Signature
Γ: Theory
a, b, c, d, e, f: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
wfd: well_formed d
wfe: well_formed e
wff: well_formed f
H: Γ ⊢ a ---> b ---> c ---> d ---> e ---> f
Γ Ⱶ
 a and b and c and d and e ---> f

  {
Σ: Signature
Γ: Theory
a, b, c, d, e, f: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
wfd: well_formed d
wfe: well_formed e
wff: well_formed f
H: Γ ⊢ a ---> b ---> c ---> d ---> e ---> f
well_formed (a and b and c and d and e ---> f) wf_auto2. }
Σ: Signature
Γ: Theory
a, b, c, d, e, f: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
wfd: well_formed d
wfe: well_formed e
wff: well_formed f
H: Γ ⊢ a ---> b ---> c ---> d ---> e ---> f
Γ Ⱶ
 a and b and c and d and e ---> f

  mlIntro "H1".
Σ: Signature
Γ: Theory
a, b, c, d, e, f: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
wfd: well_formed d
wfe: well_formed e
wff: well_formed f
H: Γ ⊢ a ---> b ---> c ---> d ---> e ---> f
Γ Ⱶ
"H1" ∶ a and b and c and d and e,
--------------------------------------
f

  mlApplyMetaGeneralized H.
Σ: Signature
Γ: Theory
a, b, c, d, e, f: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
wfd: well_formed d
wfe: well_formed e
wff: well_formed f
H: Γ ⊢ a ---> b ---> c ---> d ---> e ---> f
Γ Ⱶ
"H1" ∶ a and b and c and d and e,
--------------------------------------
a and b and c and d and e

  mlExact "H1".
Defined.

Section FOL_helpers.

  Context {Σ : Signature}.

  Lemma Framing_left (Γ : Theory) (ϕ₁ ϕ₂ ψ : Pattern) (i : ProofInfo)
    (wfψ : well_formed ψ)
    {pile : ProofInfoLe ((ExGen := ∅, SVSubst := ∅, KT := false, FP := {[(exist _ ψ wfψ)]})) i}
    :
    Γ ⊢i ϕ₁ ---> ϕ₂ using i ->
    Γ ⊢i ϕ₁ $ ψ ---> ϕ₂ $ ψ using i.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
Γ ⊢i ϕ₁ ---> ϕ₂ using i
→ Γ ⊢i ϕ₁ $ ψ ---> ϕ₂ $ ψ using i
  Proof.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
Γ ⊢i ϕ₁ ---> ϕ₂ using i
→ Γ ⊢i ϕ₁ $ ψ ---> ϕ₂ $ ψ using i
    intros [pf Hpf].
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
Γ ⊢i ϕ₁ $ ψ ---> ϕ₂ $ ψ using i
    unshelve (eexists).
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ML_proof_system Γ (ϕ₁ $ ψ ---> ϕ₂ $ ψ)
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ProofInfoMeaning Γ (ϕ₁ $ ψ ---> ϕ₂ $ ψ) ?pf i
    {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ML_proof_system Γ (ϕ₁ $ ψ ---> ϕ₂ $ ψ)
      apply ProofSystem.Framing_left.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
well_formed ψ
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ML_proof_system Γ (ϕ₁ ---> ϕ₂)
      {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
well_formed ψ exact wfψ. }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ML_proof_system Γ (ϕ₁ ---> ϕ₂)
      exact pf.
    }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ProofInfoMeaning Γ (ϕ₁ $ ψ ---> ϕ₂ $ ψ)
  (Framing_left Γ ϕ₁ ϕ₂ ψ wfψ pf) i
    {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ProofInfoMeaning Γ (ϕ₁ $ ψ ---> ϕ₂ $ ψ)
  (Framing_left Γ ϕ₁ ϕ₂ ψ wfψ pf) i
      destruct Hpf as [Hpf1 Hpf2 Hpf3 Hpf4].
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
ProofInfoMeaning Γ (ϕ₁ $ ψ ---> ϕ₂ $ ψ)
  (Framing_left Γ ϕ₁ ϕ₂ ψ wfψ pf) i
      constructor; simpl.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
uses_kt pf ==> pi_uses_kt i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
      {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
        assumption.
      }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
uses_kt pf ==> pi_uses_kt i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
      {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
        assumption.
      }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
uses_kt pf ==> pi_uses_kt i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
      {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
uses_kt pf ==> pi_uses_kt i
        assumption.
      }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
      {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
        destruct i.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]})
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ ProofSystem.pi_generalized_evars
    (ExGen := pi_generalized_evars,
     SVSubst := pi_substituted_svars,
     KT := pi_uses_kt,
     FP := pi_framing_patterns)
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ ProofSystem.pi_substituted_svars
    (ExGen := pi_generalized_evars,
     SVSubst := pi_substituted_svars,
     KT := pi_uses_kt,
     FP := pi_framing_patterns)
Hpf3: uses_kt pf ==>
ProofSystem.pi_uses_kt
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ ProofSystem.pi_framing_patterns
    (ExGen := pi_generalized_evars,
     SVSubst := pi_substituted_svars,
     KT := pi_uses_kt,
     FP := pi_framing_patterns)
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ ProofSystem.pi_framing_patterns
    (ExGen := pi_generalized_evars,
     SVSubst := pi_substituted_svars,
     KT := pi_uses_kt, FP := pi_framing_patterns)
        simpl in *.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]})
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
        apply pile_evs_svs_kt_back in pile.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pile: ∅ ⊆ pi_generalized_evars
∧ ∅ ⊆ pi_substituted_svars
  ∧ false ==> pi_uses_kt
    ∧ {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
        destruct_and!.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H: ∅ ⊆ pi_generalized_evars
H1: ∅ ⊆ pi_substituted_svars
H0: false ==> pi_uses_kt
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
        (*
        destruct pi_framing_patterns0.
        { exfalso. set_solver. } *)
        clear H H1 H0.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
        rewrite elem_of_subseteq.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
∀ x : wfPattern,
  x
  ∈ gset_to_coGset
      ({[ψ ↾ wfψ]}
       ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
  → x ∈ pi_framing_patterns
        intros p0 Hp0.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0
∈ gset_to_coGset
    ({[ψ ↾ wfψ]}
     ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
p0 ∈ pi_framing_patterns
        rewrite elem_of_gset_to_coGset in Hp0.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0
∈ {[ψ ↾ wfψ]}
  ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf
p0 ∈ pi_framing_patterns
        rewrite elem_of_union in Hp0.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
∨ p0 ∈ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf
p0 ∈ pi_framing_patterns
        destruct Hp0 as [Hp0|Hp0].
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
p0 ∈ pi_framing_patterns
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf
p0 ∈ pi_framing_patterns
        {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
p0 ∈ pi_framing_patterns
          subst.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
p0 ∈ pi_framing_patterns
          remember ((@sval) Pattern (λ p : Pattern, well_formed p = true)) as F.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
p0 ∈ pi_framing_patterns
          rewrite elem_of_subseteq in H3.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ∀ x : wfPattern,
  x ∈ {[ψ ↾ wfψ]} → x ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns
          specialize (H3 (exist _ ψ wfψ)).
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ψ ↾ wfψ ∈ {[ψ ↾ wfψ]}
→ ψ ↾ wfψ ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns
          feed specialize H3.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ψ ↾ wfψ ∈ {[ψ ↾ wfψ]}
→ ψ ↾ wfψ ∈ pi_framing_patterns
ψ ↾ wfψ ∈ {[ψ ↾ wfψ]}
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ψ ↾ wfψ ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns
          {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ψ ↾ wfψ ∈ {[ψ ↾ wfψ]}
→ ψ ↾ wfψ ∈ pi_framing_patterns
ψ ↾ wfψ ∈ {[ψ ↾ wfψ]}
            clear.
Σ: Signature
ψ: Pattern
wfψ: well_formed ψ
ψ ↾ wfψ ∈ {[ψ ↾ wfψ]} set_solver.
          }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ψ ↾ wfψ ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns
          subst F.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
H3: ψ ↾ wfψ ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns
          clear -Hp0 H3.
Σ: Signature
ψ: Pattern
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
H3: ψ ↾ wfψ ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns
          set_solver.
        }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf
p0 ∈ pi_framing_patterns
        {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf
p0 ∈ pi_framing_patterns
          set_solver.
        }
      }
    }
  Defined.

  Lemma Framing_right (Γ : Theory) (ϕ₁ ϕ₂ ψ : Pattern) (i : ProofInfo)
    (wfψ : well_formed ψ)
    {pile : ProofInfoLe ((ExGen := ∅, SVSubst := ∅, KT := false, FP := {[(exist _ ψ wfψ)]})) i}
    :
    Γ ⊢i ϕ₁ ---> ϕ₂ using i ->
    Γ ⊢i ψ $ ϕ₁ ---> ψ $ ϕ₂ using i.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
Γ ⊢i ϕ₁ ---> ϕ₂ using i
→ Γ ⊢i ψ $ ϕ₁ ---> ψ $ ϕ₂ using i
  Proof.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
Γ ⊢i ϕ₁ ---> ϕ₂ using i
→ Γ ⊢i ψ $ ϕ₁ ---> ψ $ ϕ₂ using i
    intros [pf Hpf].
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
Γ ⊢i ψ $ ϕ₁ ---> ψ $ ϕ₂ using i
    unshelve (eexists).
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ML_proof_system Γ (ψ $ ϕ₁ ---> ψ $ ϕ₂)
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ProofInfoMeaning Γ (ψ $ ϕ₁ ---> ψ $ ϕ₂) ?pf i
    {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ML_proof_system Γ (ψ $ ϕ₁ ---> ψ $ ϕ₂)
      apply ProofSystem.Framing_right.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
well_formed ψ
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ML_proof_system Γ (ϕ₁ ---> ϕ₂)
      {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
well_formed ψ exact wfψ. }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ML_proof_system Γ (ϕ₁ ---> ϕ₂)
      exact pf.
    }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ProofInfoMeaning Γ (ψ $ ϕ₁ ---> ψ $ ϕ₂)
  (Framing_right Γ ϕ₁ ϕ₂ ψ wfψ pf) i
    {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf: ProofInfoMeaning Γ (ϕ₁ ---> ϕ₂) pf i
ProofInfoMeaning Γ (ψ $ ϕ₁ ---> ψ $ ϕ₂)
  (Framing_right Γ ϕ₁ ϕ₂ ψ wfψ pf) i
      destruct Hpf as [Hpf1 Hpf2 Hpf3 Hpf4].
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
ProofInfoMeaning Γ (ψ $ ϕ₁ ---> ψ $ ϕ₂)
  (Framing_right Γ ϕ₁ ϕ₂ ψ wfψ pf) i
      constructor; simpl.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
uses_kt pf ==> pi_uses_kt i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
      {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
        assumption.
      }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
uses_kt pf ==> pi_uses_kt i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
      {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
        assumption.
      }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
uses_kt pf ==> pi_uses_kt i
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
      {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
uses_kt pf ==> pi_uses_kt i
        assumption.
      }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
      {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
i: ProofInfo
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]}) i
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars i
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars i
Hpf3: uses_kt pf ==> pi_uses_kt i
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns i
        destruct i.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]})
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ ProofSystem.pi_generalized_evars
    (ExGen := pi_generalized_evars,
     SVSubst := pi_substituted_svars,
     KT := pi_uses_kt,
     FP := pi_framing_patterns)
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ ProofSystem.pi_substituted_svars
    (ExGen := pi_generalized_evars,
     SVSubst := pi_substituted_svars,
     KT := pi_uses_kt,
     FP := pi_framing_patterns)
Hpf3: uses_kt pf ==>
ProofSystem.pi_uses_kt
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ ProofSystem.pi_framing_patterns
    (ExGen := pi_generalized_evars,
     SVSubst := pi_substituted_svars,
     KT := pi_uses_kt,
     FP := pi_framing_patterns)
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ ProofSystem.pi_framing_patterns
    (ExGen := pi_generalized_evars,
     SVSubst := pi_substituted_svars,
     KT := pi_uses_kt, FP := pi_framing_patterns)
        simpl in *.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ ↾ wfψ]})
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
        apply pile_evs_svs_kt_back in pile.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pile: ∅ ⊆ pi_generalized_evars
∧ ∅ ⊆ pi_substituted_svars
  ∧ false ==> pi_uses_kt
    ∧ {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
        destruct_and!.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H: ∅ ⊆ pi_generalized_evars
H1: ∅ ⊆ pi_substituted_svars
H0: false ==> pi_uses_kt
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
        clear H H1 H0.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
gset_to_coGset
  ({[ψ ↾ wfψ]} ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
        rewrite elem_of_subseteq.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
∀ x : wfPattern,
  x
  ∈ gset_to_coGset
      ({[ψ ↾ wfψ]}
       ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
  → x ∈ pi_framing_patterns
        intros p0 Hp0.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0
∈ gset_to_coGset
    ({[ψ ↾ wfψ]}
     ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
p0 ∈ pi_framing_patterns
        rewrite elem_of_gset_to_coGset in Hp0.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0
∈ {[ψ ↾ wfψ]}
  ∪ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf
p0 ∈ pi_framing_patterns
        rewrite elem_of_union in Hp0.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
∨ p0 ∈ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf
p0 ∈ pi_framing_patterns
        destruct Hp0 as [Hp0|Hp0].
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
p0 ∈ pi_framing_patterns
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf
p0 ∈ pi_framing_patterns
        {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
p0 ∈ pi_framing_patterns
          subst.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
p0 ∈ pi_framing_patterns
          remember ((@sval) Pattern (λ p : Pattern, well_formed p = true)) as F.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
p0 ∈ pi_framing_patterns
          rewrite elem_of_subseteq in H3.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ∀ x : wfPattern,
  x ∈ {[ψ ↾ wfψ]} → x ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns
          specialize (H3 (exist _ ψ wfψ)).
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ψ ↾ wfψ ∈ {[ψ ↾ wfψ]}
→ ψ ↾ wfψ ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns
          feed specialize H3.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ψ ↾ wfψ ∈ {[ψ ↾ wfψ]}
→ ψ ↾ wfψ ∈ pi_framing_patterns
ψ ↾ wfψ ∈ {[ψ ↾ wfψ]}
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ψ ↾ wfψ ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns
          {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ψ ↾ wfψ ∈ {[ψ ↾ wfψ]}
→ ψ ↾ wfψ ∈ pi_framing_patterns
ψ ↾ wfψ ∈ {[ψ ↾ wfψ]}
            clear.
Σ: Signature
ψ: Pattern
wfψ: well_formed ψ
ψ ↾ wfψ ∈ {[ψ ↾ wfψ]} set_solver.
          }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
F: {p : Pattern | well_formed p = true} → Pattern
HeqF: F =
(@sval) Pattern
  (λ p : Pattern, well_formed p = true)
H3: ψ ↾ wfψ ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns
          subst F.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
H3: ψ ↾ wfψ ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns
          clear -Hp0 H3.
Σ: Signature
ψ: Pattern
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
p0: wfPattern
Hp0: p0 ∈ {[ψ ↾ wfψ]}
H3: ψ ↾ wfψ ∈ pi_framing_patterns
p0 ∈ pi_framing_patterns set_solver.
        }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf
p0 ∈ pi_framing_patterns
        {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
wfψ: well_formed ψ
H3: {[ψ ↾ wfψ]} ⊆ pi_framing_patterns
pf: ML_proof_system Γ (ϕ₁ ---> ϕ₂)
Hpf1: gset_to_coGset
  (uses_of_ex_gen Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_generalized_evars
Hpf2: gset_to_coGset
  (uses_of_svar_subst Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_substituted_svars
Hpf3: uses_kt pf ==> pi_uses_kt
Hpf4: gset_to_coGset
  (framing_patterns Γ (ϕ₁ ---> ϕ₂) pf)
⊆ pi_framing_patterns
p0: wfPattern
Hp0: p0 ∈ framing_patterns Γ (ϕ₁ ---> ϕ₂) pf
p0 ∈ pi_framing_patterns
          set_solver.
        }
      }
    }
  Defined.

  Lemma Prop_bott_left (Γ : Theory) (ϕ : Pattern) :
    well_formed ϕ ->
    Γ ⊢i ⊥ $ ϕ ---> ⊥ using BasicReasoning.
Σ: Signature
Γ: Theory
ϕ: Pattern
well_formed ϕ → Γ ⊢i ⊥ $ ϕ ---> ⊥ using BasicReasoning
  Proof.
Σ: Signature
Γ: Theory
ϕ: Pattern
well_formed ϕ → Γ ⊢i ⊥ $ ϕ ---> ⊥ using BasicReasoning
    intros wfϕ.
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
Γ ⊢i ⊥ $ ϕ ---> ⊥ using BasicReasoning
    unshelve (eexists).
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
ML_proof_system Γ (⊥ $ ϕ ---> ⊥)
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
ProofInfoMeaning Γ (⊥ $ ϕ ---> ⊥) ?pf BasicReasoning
    {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
ML_proof_system Γ (⊥ $ ϕ ---> ⊥)
      apply ProofSystem.Prop_bott_left.
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
well_formed ϕ exact wfϕ.
    }
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
ProofInfoMeaning Γ (⊥ $ ϕ ---> ⊥)
  (Prop_bott_left Γ ϕ wfϕ) BasicReasoning
    {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
ProofInfoMeaning Γ (⊥ $ ϕ ---> ⊥)
  (Prop_bott_left Γ ϕ wfϕ) BasicReasoning
      constructor; simpl.
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
true
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
      {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
        set_solver.
      }
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
true
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
      {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
        set_solver.
      }
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
true
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
      {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
true
        reflexivity.
      }
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
      {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
        apply reflexivity.
      }
    }
  Defined.

  Lemma Prop_bott_right (Γ : Theory) (ϕ : Pattern) :
    well_formed ϕ ->
    Γ ⊢i ϕ $ ⊥ ---> ⊥ using BasicReasoning.
Σ: Signature
Γ: Theory
ϕ: Pattern
well_formed ϕ → Γ ⊢i ϕ $ ⊥ ---> ⊥ using BasicReasoning
  Proof.
Σ: Signature
Γ: Theory
ϕ: Pattern
well_formed ϕ → Γ ⊢i ϕ $ ⊥ ---> ⊥ using BasicReasoning
    intros wfϕ.
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
Γ ⊢i ϕ $ ⊥ ---> ⊥ using BasicReasoning
    unshelve (eexists).
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
ML_proof_system Γ (ϕ $ ⊥ ---> ⊥)
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
ProofInfoMeaning Γ (ϕ $ ⊥ ---> ⊥) ?pf BasicReasoning
    {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
ML_proof_system Γ (ϕ $ ⊥ ---> ⊥)
      apply ProofSystem.Prop_bott_right.
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
well_formed ϕ exact wfϕ.
    }
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
ProofInfoMeaning Γ (ϕ $ ⊥ ---> ⊥)
  (Prop_bott_right Γ ϕ wfϕ) BasicReasoning
    {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
ProofInfoMeaning Γ (ϕ $ ⊥ ---> ⊥)
  (Prop_bott_right Γ ϕ wfϕ) BasicReasoning
      constructor; simpl.
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
true
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
      {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
        set_solver.
      }
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
true
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
      {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
        set_solver.
      }
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
true
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
      {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
true
        reflexivity.
      }
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
      {
Σ: Signature
Γ: Theory
ϕ: Pattern
wfϕ: well_formed ϕ
gset_to_coGset ∅ ⊆ ∅
        apply reflexivity.
      }
    }
  Defined.

  Arguments Prop_bott_left _ (_%ml) _ : clear implicits.
  Arguments Prop_bott_right _ (_%ml) _ : clear implicits.

  Fixpoint frames_of_AC (C : Application_context)
    : coWfpSet :=
  match C with
  | box => ∅
  | ctx_app_l C' p wfp => {[(exist _ p wfp)]} ∪ (frames_of_AC C')
  | ctx_app_r p C' wfp => {[(exist _ p wfp)]} ∪ (frames_of_AC C')
  end.


  (* TODO rename into Prop_bot_ctx *)
  Lemma Prop_bot (Γ : Theory) (C : Application_context) :
    Γ ⊢i ((subst_ctx C patt_bott) ---> patt_bott)
    using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC C).
Σ: Signature
Γ: Theory
C: Application_context
Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC C)
  Proof.
Σ: Signature
Γ: Theory
C: Application_context
Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC C)
    remember (frames_of_AC C) as foC.
Σ: Signature
Γ: Theory
C: Application_context
foC: coWfpSet
HeqfoC: foC = frames_of_AC C
Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC)
    move: foC HeqfoC.
Σ: Signature
Γ: Theory
C: Application_context
∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅, KT := false,
           FP := foC)
    induction C; intros foC HeqfoC; simpl in *.
Σ: Signature
Γ: Theory
foC: coWfpSet
HeqfoC: foC = ∅
Γ ⊢i ⊥ ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i C.[□↦⊥] $ p ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i p $ C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC)
    -
Σ: Signature
Γ: Theory
foC: coWfpSet
HeqfoC: foC = ∅
Γ ⊢i ⊥ ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC) apply useBasicReasoning.
Σ: Signature
Γ: Theory
foC: coWfpSet
HeqfoC: foC = ∅
Γ ⊢i ⊥ ---> ⊥ using BasicReasoning
      apply bot_elim.
Σ: Signature
Γ: Theory
foC: coWfpSet
HeqfoC: foC = ∅
well_formed ⊥
      wf_auto2.
    -
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i C.[□↦⊥] $ p ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC) eapply syllogism_meta.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed (C.[□↦⊥] $ p)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed ?B
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed ⊥
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i C.[□↦⊥] $ p ---> ?B
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i ?B ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC)
      5: {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i ?B ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC) apply useBasicReasoning.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i ?B ---> ⊥ using BasicReasoning apply (Prop_bott_left Γ p ltac:(wf_auto2)). }
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed (C.[□↦⊥] $ p)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed (⊥ $ p)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed ⊥
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i C.[□↦⊥] $ p ---> ⊥ $ p
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC)
      4: {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i C.[□↦⊥] $ p ---> ⊥ $ p
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC) simpl.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i C.[□↦⊥] $ p ---> ⊥ $ p
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC) subst foC.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i C.[□↦⊥] $ p ---> ⊥ $ p
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p ↾ Prf]} ∪ frames_of_AC C)  eapply useGenericReasoning.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
   FP := ?fp)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i C.[□↦⊥] $ p ---> ⊥ $ p
using (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
       FP := ?fp)
           2: eapply Framing_left with (wfψ := Prf).
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
   FP := ?fp)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]})
  (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
   FP := ?fp)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
       FP := ?fp)
           1: apply pile_refl.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p ↾ Prf]} ∪ frames_of_AC C)
           1: {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
            eapply pile_evs_svs_kt.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
            {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅ apply reflexivity. }
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
            {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅ apply reflexivity. }
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
            {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false reflexivity. }
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
            {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C clear.
Σ: Signature
C: Application_context
p: Pattern
Prf: well_formed p
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C set_solver. }
           }
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p ↾ Prf]} ∪ frames_of_AC C)
           eapply useGenericReasoning.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
   FP := ?fp)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
       FP := ?fp)
           2: apply IHC.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false, FP := ?fp)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
?fp = frames_of_AC C
           2: reflexivity.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
           apply pile_evs_svs_kt.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
           {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅ apply reflexivity. }
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
           {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅ apply reflexivity. }
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
           {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false reflexivity. }
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
           {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C clear.
Σ: Signature
C: Application_context
p: Pattern
Prf: well_formed p
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C set_solver. }
      }
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed (C.[□↦⊥] $ p)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed (⊥ $ p)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed ⊥
      all: wf_auto2.
       -
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i p $ C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC) eapply syllogism_meta.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed (p $ C.[□↦⊥])
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed ?B
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed ⊥
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i p $ C.[□↦⊥] ---> ?B
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i ?B ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC)
           5: {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i ?B ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC) apply useBasicReasoning.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i ?B ---> ⊥ using BasicReasoning apply (Prop_bott_right Γ p ltac:(wf_auto2)). }
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed (p $ C.[□↦⊥])
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed (p $ ⊥)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed ⊥
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i p $ C.[□↦⊥] ---> p $ ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC)
           4: {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i p $ C.[□↦⊥] ---> p $ ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC) simpl.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
Γ ⊢i p $ C.[□↦⊥] ---> p $ ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := foC) subst foC.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i p $ C.[□↦⊥] ---> p $ ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p ↾ Prf]} ∪ frames_of_AC C)  eapply useGenericReasoning.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
   FP := ?fp)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i p $ C.[□↦⊥] ---> p $ ⊥
using (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
       FP := ?fp)
                2: eapply Framing_right with (wfψ := Prf).
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
   FP := ?fp)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]})
  (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
   FP := ?fp)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
       FP := ?fp)
                1: apply pile_refl.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p ↾ Prf]} ∪ frames_of_AC C)
                1: {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
                 eapply pile_evs_svs_kt.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
                 {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅ apply reflexivity. }
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
                 {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅ apply reflexivity. }
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
                 {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false reflexivity. }
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
                 {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C clear.
Σ: Signature
p: Pattern
C: Application_context
Prf: well_formed p
{[p ↾ Prf]} ⊆ {[p ↾ Prf]} ∪ frames_of_AC C set_solver. }
                }
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p ↾ Prf]} ∪ frames_of_AC C)
                eapply useGenericReasoning.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
   FP := ?fp)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
       FP := ?fp)
                2: apply IHC.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false, FP := ?fp)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
?fp = frames_of_AC C
                2: reflexivity.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
                apply pile_evs_svs_kt.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
                {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅ apply reflexivity. }
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
                {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
∅ ⊆ ∅ apply reflexivity. }
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
                {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
false ==> false reflexivity. }
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C
                {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C clear.
Σ: Signature
p: Pattern
C: Application_context
Prf: well_formed p
frames_of_AC C ⊆ {[p ↾ Prf]} ∪ frames_of_AC C set_solver. }
           }
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
foC = frames_of_AC C
→ Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed (p $ C.[□↦⊥])
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed (p $ ⊥)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
IHC: ∀ foC : coWfpSet,
  foC = frames_of_AC C
  → Γ ⊢i C.[□↦⊥] ---> ⊥
    using (ExGen := ∅, SVSubst := ∅,
           KT := false, FP := foC)
foC: coWfpSet
HeqfoC: foC = {[p ↾ Prf]} ∪ frames_of_AC C
well_formed ⊥
           all: wf_auto2.
  Defined.

  Lemma Framing (Γ : Theory) (C : Application_context) (A B : Pattern) (i : ProofInfo)
    {pile : ProofInfoLe
     ((ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC C))
     i
    }
    :
    Γ ⊢i (A ---> B) using i ->
    Γ ⊢i ((subst_ctx C A) ---> (subst_ctx C B)) using i.
Σ: Signature
Γ: Theory
C: Application_context
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
Γ ⊢i A ---> B using i
→ Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
  Proof.
Σ: Signature
Γ: Theory
C: Application_context
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
Γ ⊢i A ---> B using i
→ Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
    intros H.
Σ: Signature
Γ: Theory
C: Application_context
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
H: Γ ⊢i A ---> B using i
Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
    pose proof H as [pf _].
Σ: Signature
Γ: Theory
C: Application_context
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
H: Γ ⊢i A ---> B using i
pf: ML_proof_system Γ (A ---> B)
Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
    pose proof (HWF := proved_impl_wf _ _ pf).
Σ: Signature
Γ: Theory
C: Application_context
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
H: Γ ⊢i A ---> B using i
pf: ML_proof_system Γ (A ---> B)
HWF: well_formed (A ---> B)
Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
    assert (wfA: well_formed A) by wf_auto2.
Σ: Signature
Γ: Theory
C: Application_context
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
H: Γ ⊢i A ---> B using i
pf: ML_proof_system Γ (A ---> B)
HWF: well_formed (A ---> B)
wfA: well_formed A
Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
    assert (wfB: well_formed B) by wf_auto2.
Σ: Signature
Γ: Theory
C: Application_context
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
H: Γ ⊢i A ---> B using i
pf: ML_proof_system Γ (A ---> B)
HWF: well_formed (A ---> B)
wfA: well_formed A
wfB: well_formed B
Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
    clear pf HWF.
Σ: Signature
Γ: Theory
C: Application_context
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
H: Γ ⊢i A ---> B using i
wfA: well_formed A
wfB: well_formed B
Γ ⊢i C.[□↦A] ---> C.[□↦B] using i

    move: wfA wfB H.
Σ: Signature
Γ: Theory
C: Application_context
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
well_formed A
→ well_formed B
  → Γ ⊢i A ---> B using i
    → Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
    induction C; intros WFA WFB H; simpl in *.
Σ: Signature
Γ: Theory
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := ∅) i
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B using i
Γ ⊢i A ---> B using i
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C) i
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B using i
      → Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B using i
Γ ⊢i C.[□↦A] $ p ---> C.[□↦B] $ p using i
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C) i
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B using i
      → Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B using i
Γ ⊢i p $ C.[□↦A] ---> p $ C.[□↦B] using i
    -
Σ: Signature
Γ: Theory
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := ∅) i
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B using i
Γ ⊢i A ---> B using i exact H.
    -
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C) i
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B using i
      → Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B using i
Γ ⊢i C.[□↦A] $ p ---> C.[□↦B] $ p using i destruct i.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i C.[□↦A] $ p ---> C.[□↦B] $ p
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      apply pile_evs_svs_kt_back in pile.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
pile: ∅ ⊆ pi_generalized_evars
∧ ∅ ⊆ pi_substituted_svars
  ∧ false ==> pi_uses_kt
    ∧ {[p ↾ Prf]} ∪ frames_of_AC C
      ⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i C.[□↦A] $ p ---> C.[□↦B] $ p
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      destruct_and!.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: false ==> pi_uses_kt
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i C.[□↦A] $ p ---> C.[□↦B] $ p
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns) simpl in *.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i C.[□↦A] $ p ---> C.[□↦B] $ p
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      apply Framing_left with (wfψ := Prf).
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]})
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars, KT := pi_uses_kt,
   FP := pi_framing_patterns)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i C.[□↦A] ---> C.[□↦B]
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      {
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]})
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars, KT := pi_uses_kt,
   FP := pi_framing_patterns)
        apply pile_evs_svs_kt; try assumption.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
{[p ↾ Prf]} ⊆ pi_framing_patterns
        set_solver.
      }
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i C.[□↦A] ---> C.[□↦B]
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      apply IHC.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars, KT := pi_uses_kt,
   FP := pi_framing_patterns)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
well_formed A
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
well_formed B
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      2,3: assumption.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars, KT := pi_uses_kt,
   FP := pi_framing_patterns)
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      2: exact H.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars, KT := pi_uses_kt,
   FP := pi_framing_patterns)
      apply pile_evs_svs_kt; try assumption.
Σ: Signature
Γ: Theory
C: Application_context
p: Pattern
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
frames_of_AC C ⊆ pi_framing_patterns
      set_solver.
    -
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C) i
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B using i
      → Γ ⊢i C.[□↦A] ---> C.[□↦B] using i
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B using i
Γ ⊢i p $ C.[□↦A] ---> p $ C.[□↦B] using i destruct i.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]} ∪ frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i p $ C.[□↦A] ---> p $ C.[□↦B]
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      apply pile_evs_svs_kt_back in pile.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
pile: ∅ ⊆ pi_generalized_evars
∧ ∅ ⊆ pi_substituted_svars
  ∧ false ==> pi_uses_kt
    ∧ {[p ↾ Prf]} ∪ frames_of_AC C
      ⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i p $ C.[□↦A] ---> p $ C.[□↦B]
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      destruct_and!.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: false ==> pi_uses_kt
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i p $ C.[□↦A] ---> p $ C.[□↦B]
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns) simpl in *.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i p $ C.[□↦A] ---> p $ C.[□↦B]
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      apply Framing_right with (wfψ := Prf).
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]})
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars, KT := pi_uses_kt,
   FP := pi_framing_patterns)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i C.[□↦A] ---> C.[□↦B]
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      {
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p ↾ Prf]})
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars, KT := pi_uses_kt,
   FP := pi_framing_patterns)
        apply pile_evs_svs_kt; try assumption.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
{[p ↾ Prf]} ⊆ pi_framing_patterns
        set_solver.
      }
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i C.[□↦A] ---> C.[□↦B]
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      apply IHC.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars, KT := pi_uses_kt,
   FP := pi_framing_patterns)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
well_formed A
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
well_formed B
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      2,3: assumption.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars, KT := pi_uses_kt,
   FP := pi_framing_patterns)
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
      2: exact H.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars, KT := pi_uses_kt,
   FP := pi_framing_patterns)
      apply pile_evs_svs_kt; try assumption.
Σ: Signature
Γ: Theory
p: Pattern
C: Application_context
Prf: well_formed p
A, B: Pattern
pi_generalized_evars: coEVarSet
pi_substituted_svars: coSVarSet
pi_uses_kt: bool
pi_framing_patterns: coWfpSet
H0: ∅ ⊆ pi_generalized_evars
H2: ∅ ⊆ pi_substituted_svars
H1: true
H4: {[p ↾ Prf]} ∪ frames_of_AC C
⊆ pi_framing_patterns
IHC: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C)
  (ExGen := pi_generalized_evars,
   SVSubst := pi_substituted_svars,
   KT := pi_uses_kt, FP := pi_framing_patterns)
→ well_formed A
  → well_formed B
    → Γ ⊢i A ---> B
      using (ExGen := pi_generalized_evars,
             SVSubst := pi_substituted_svars,
             KT := pi_uses_kt,
             FP := pi_framing_patterns)
      → Γ ⊢i C.[□↦A] ---> C.[□↦B]
        using (ExGen := pi_generalized_evars,
               SVSubst := pi_substituted_svars,
               KT := pi_uses_kt,
               FP := pi_framing_patterns)
WFA: well_formed A
WFB: well_formed B
H: Γ ⊢i A ---> B
using (ExGen := pi_generalized_evars,
       SVSubst := pi_substituted_svars,
       KT := pi_uses_kt, FP := pi_framing_patterns)
frames_of_AC C ⊆ pi_framing_patterns
      set_solver.
  Defined.

  Lemma A_implies_not_not_A_ctx (Γ : Theory) (A : Pattern) (C : Application_context)
    (i : ProofInfo) {pile : ProofInfoLe ((ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC C)) i}
    :
    well_formed A ->
    Γ ⊢i A using i ->
    Γ ⊢i (! (subst_ctx C ( !A ))) using i.
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
well_formed A
→ Γ ⊢i A using i → Γ ⊢i ! C.[□↦! A] using i
  Proof.
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
well_formed A
→ Γ ⊢i A using i → Γ ⊢i ! C.[□↦! A] using i
    intros WFA H.
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
Γ ⊢i ! C.[□↦! A] using i

    epose proof (ANNA := A_implies_not_not_A_alt Γ _ i _ H).
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
ANNA: Γ ⊢i ! ! A using i
Γ ⊢i ! C.[□↦! A] using i
    replace (! (! A)) with ((! A) ---> Bot) in ANNA by reflexivity.
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
ANNA: Γ ⊢i ! A ---> ⊥ using i
Γ ⊢i ! C.[□↦! A] using i
    epose proof (EF := Framing _ C (! A) Bot _ ANNA).
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
ANNA: Γ ⊢i ! A ---> ⊥ using i
EF: Γ ⊢i C.[□↦! A] ---> C.[□↦⊥] using i
Γ ⊢i ! C.[□↦! A] using i
    epose proof (PB := Prop_bot Γ C).
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
ANNA: Γ ⊢i ! A ---> ⊥ using i
EF: Γ ⊢i C.[□↦! A] ---> C.[□↦⊥] using i
PB: Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC C)
Γ ⊢i ! C.[□↦! A] using i
    apply liftPi with (i₂ := i) in PB.
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
ANNA: Γ ⊢i ! A ---> ⊥ using i
EF: Γ ⊢i C.[□↦! A] ---> C.[□↦⊥] using i
PB: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
Γ ⊢i ! C.[□↦! A] using i
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
ANNA: Γ ⊢i ! A ---> ⊥ using i
EF: Γ ⊢i C.[□↦! A] ---> C.[□↦⊥] using i
PB: Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC C)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i 2: apply _.
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
ANNA: Γ ⊢i ! A ---> ⊥ using i
EF: Γ ⊢i C.[□↦! A] ---> C.[□↦⊥] using i
PB: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
Γ ⊢i ! C.[□↦! A] using i
    epose (TRANS := syllogism_meta _ _ _ EF PB).
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
ANNA: Γ ⊢i ! A ---> ⊥ using i
EF: Γ ⊢i C.[□↦! A] ---> C.[□↦⊥] using i
PB: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
TRANS:= syllogism_meta ?Goal0 ?Goal1 ?Goal2 EF PB: Γ ⊢i C.[□↦! A] ---> ⊥ using i
Γ ⊢i ! C.[□↦! A] using i
    apply TRANS.

    Unshelve.
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
well_formed A
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
ANNA: Γ ⊢i ! A ---> ⊥ using i
EF: Γ ⊢i C.[□↦! A] ---> C.[□↦⊥] using i
PB: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
well_formed C.[□↦! A]
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
ANNA: Γ ⊢i ! A ---> ⊥ using i
EF: Γ ⊢i C.[□↦! A] ---> C.[□↦⊥] using i
PB: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
well_formed C.[□↦⊥]
Σ: Signature
Γ: Theory
A: Pattern
C: Application_context
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A using i
ANNA: Γ ⊢i ! A ---> ⊥ using i
EF: Γ ⊢i C.[□↦! A] ---> C.[□↦⊥] using i
PB: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
well_formed ⊥
    all: wf_auto2.
  Defined.

  Lemma ctx_bot_prop (Γ : Theory) (C : Application_context) (A : Pattern) 
    (i : ProofInfo)
    {pile : ProofInfoLe ((ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC C)) i}
  :
    well_formed A ->
    Γ ⊢i (A ---> Bot) using i ->
    Γ ⊢i (subst_ctx C A ---> Bot) using i.
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
well_formed A
→ Γ ⊢i A ---> ⊥ using i → Γ ⊢i C.[□↦A] ---> ⊥ using i
  Proof.
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
well_formed A
→ Γ ⊢i A ---> ⊥ using i → Γ ⊢i C.[□↦A] ---> ⊥ using i
    intros WFA H.
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A ---> ⊥ using i
Γ ⊢i C.[□↦A] ---> ⊥ using i
    epose proof (FR := Framing Γ C A Bot _ H).
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A ---> ⊥ using i
FR: Γ ⊢i C.[□↦A] ---> C.[□↦⊥] using i
Γ ⊢i C.[□↦A] ---> ⊥ using i
    epose proof (BPR := Prop_bot Γ C).
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A ---> ⊥ using i
FR: Γ ⊢i C.[□↦A] ---> C.[□↦⊥] using i
BPR: Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC C)
Γ ⊢i C.[□↦A] ---> ⊥ using i
    apply liftPi with (i₂ := i) in BPR.
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A ---> ⊥ using i
FR: Γ ⊢i C.[□↦A] ---> C.[□↦⊥] using i
BPR: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
Γ ⊢i C.[□↦A] ---> ⊥ using i
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A ---> ⊥ using i
FR: Γ ⊢i C.[□↦A] ---> C.[□↦⊥] using i
BPR: Γ ⊢i C.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC C)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i 2: apply _.
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A ---> ⊥ using i
FR: Γ ⊢i C.[□↦A] ---> C.[□↦⊥] using i
BPR: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
Γ ⊢i C.[□↦A] ---> ⊥ using i
    epose proof (TRANS := syllogism_meta _ _ _ FR BPR).
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A ---> ⊥ using i
FR: Γ ⊢i C.[□↦A] ---> C.[□↦⊥] using i
BPR: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
TRANS: Γ ⊢i C.[□↦A] ---> ⊥ using i
Γ ⊢i C.[□↦A] ---> ⊥ using i
    exact TRANS.
    Unshelve.
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A ---> ⊥ using i
FR: Γ ⊢i C.[□↦A] ---> C.[□↦⊥] using i
BPR: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
well_formed C.[□↦A]
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A ---> ⊥ using i
FR: Γ ⊢i C.[□↦A] ---> C.[□↦⊥] using i
BPR: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
well_formed C.[□↦⊥]
Σ: Signature
Γ: Theory
C: Application_context
A: Pattern
i: ProofInfo
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC C) i
WFA: well_formed A
H: Γ ⊢i A ---> ⊥ using i
FR: Γ ⊢i C.[□↦A] ---> C.[□↦⊥] using i
BPR: Γ ⊢i C.[□↦⊥] ---> ⊥ using i
well_formed ⊥
    all: wf_auto2.
  Defined.

End FOL_helpers.

Lemma prf_prop_bott_iff {Σ : Signature} Γ AC:
  Γ ⊢i ((subst_ctx AC patt_bott) <---> patt_bott)
  using (
  (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)).
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i AC.[□↦⊥] <---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Proof.
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i AC.[□↦⊥] <---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
  apply pf_iff_split.
Σ: Signature
Γ: Theory
AC: Application_context
well_formed AC.[□↦⊥]
Σ: Signature
Γ: Theory
AC: Application_context
well_formed ⊥
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i AC.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> AC.[□↦⊥]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
  1,2: wf_auto2.
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i AC.[□↦⊥] ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> AC.[□↦⊥]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
  1: apply ctx_bot_prop.
Σ: Signature
Γ: Theory
AC: Application_context
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC AC)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
well_formed ⊥
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> AC.[□↦⊥]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
  1: apply pile_refl.
Σ: Signature
Γ: Theory
AC: Application_context
well_formed ⊥
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> AC.[□↦⊥]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
  1: wf_auto2.
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> AC.[□↦⊥]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
  {
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> ⊥
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
    useBasicReasoning.
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> ⊥ using BasicReasoning
    apply A_impl_A.
Σ: Signature
Γ: Theory
AC: Application_context
well_formed ⊥
    reflexivity.
  }
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> AC.[□↦⊥]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
  {
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> AC.[□↦⊥]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
    useBasicReasoning.
Σ: Signature
Γ: Theory
AC: Application_context
Γ ⊢i ⊥ ---> AC.[□↦⊥] using BasicReasoning
    apply bot_elim.
Σ: Signature
Γ: Theory
AC: Application_context
well_formed AC.[□↦⊥]
    wf_auto2.
  }
Defined.

Lemma Prop_disj_left {Σ : Signature} (Γ : Theory) (ϕ₁ ϕ₂ ψ : Pattern) :
  well_formed ϕ₁ ->
  well_formed ϕ₂ ->
  well_formed ψ ->
  Γ ⊢i (ϕ₁ or ϕ₂) $ ψ ---> (ϕ₁ $ ψ) or (ϕ₂ $ ψ) using BasicReasoning.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
well_formed ϕ₁
→ well_formed ϕ₂
  → well_formed ψ
    → Γ ⊢i (ϕ₁ or ϕ₂) $ ψ ---> ϕ₁ $ ψ or ϕ₂ $ ψ
      using BasicReasoning
Proof.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
well_formed ϕ₁
→ well_formed ϕ₂
  → well_formed ψ
    → Γ ⊢i (ϕ₁ or ϕ₂) $ ψ ---> ϕ₁ $ ψ or ϕ₂ $ ψ
      using BasicReasoning
  intros wfϕ₁ wfϕ₂ wfψ.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
Γ ⊢i (ϕ₁ or ϕ₂) $ ψ ---> ϕ₁ $ ψ or ϕ₂ $ ψ
using BasicReasoning
  unshelve (eexists).
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
ML_proof_system Γ
  ((ϕ₁ or ϕ₂) $ ψ ---> ϕ₁ $ ψ or ϕ₂ $ ψ)
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
ProofInfoMeaning Γ
  ((ϕ₁ or ϕ₂) $ ψ ---> ϕ₁ $ ψ or ϕ₂ $ ψ) 
  ?pf BasicReasoning
  {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
ML_proof_system Γ
  ((ϕ₁ or ϕ₂) $ ψ ---> ϕ₁ $ ψ or ϕ₂ $ ψ)
    apply Prop_disj_left; assumption.
  }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
ProofInfoMeaning Γ
  ((ϕ₁ or ϕ₂) $ ψ ---> ϕ₁ $ ψ or ϕ₂ $ ψ)
  (Prop_disj_left Γ ϕ₁ ϕ₂ ψ wfϕ₁ wfϕ₂ wfψ)
  BasicReasoning
  {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
ProofInfoMeaning Γ
  ((ϕ₁ or ϕ₂) $ ψ ---> ϕ₁ $ ψ or ϕ₂ $ ψ)
  (Prop_disj_left Γ ϕ₁ ϕ₂ ψ wfϕ₁ wfϕ₂ wfψ)
  BasicReasoning
    abstract (
      constructor; simpl;
      [(set_solver)|(set_solver)|(reflexivity)|(set_solver)]
    ).
  }
Defined.

Lemma Prop_disj_right {Σ : Signature} (Γ : Theory) (ϕ₁ ϕ₂ ψ : Pattern) :
  well_formed ϕ₁ ->
  well_formed ϕ₂ ->
  well_formed ψ ->
  Γ ⊢i ψ $ (ϕ₁ or ϕ₂)  ---> (ψ $ ϕ₁) or (ψ $ ϕ₂) using BasicReasoning.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
well_formed ϕ₁
→ well_formed ϕ₂
  → well_formed ψ
    → Γ ⊢i ψ $ (ϕ₁ or ϕ₂) ---> ψ $ ϕ₁ or ψ $ ϕ₂
      using BasicReasoning
Proof.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
well_formed ϕ₁
→ well_formed ϕ₂
  → well_formed ψ
    → Γ ⊢i ψ $ (ϕ₁ or ϕ₂) ---> ψ $ ϕ₁ or ψ $ ϕ₂
      using BasicReasoning
  intros wfϕ₁ wfϕ₂ wfψ.
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
Γ ⊢i ψ $ (ϕ₁ or ϕ₂) ---> ψ $ ϕ₁ or ψ $ ϕ₂
using BasicReasoning
  unshelve (eexists).
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
ML_proof_system Γ
  (ψ $ (ϕ₁ or ϕ₂) ---> ψ $ ϕ₁ or ψ $ ϕ₂)
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
ProofInfoMeaning Γ
  (ψ $ (ϕ₁ or ϕ₂) ---> ψ $ ϕ₁ or ψ $ ϕ₂) 
  ?pf BasicReasoning
  {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
ML_proof_system Γ
  (ψ $ (ϕ₁ or ϕ₂) ---> ψ $ ϕ₁ or ψ $ ϕ₂)
    apply Prop_disj_right; assumption.
  }
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
ProofInfoMeaning Γ
  (ψ $ (ϕ₁ or ϕ₂) ---> ψ $ ϕ₁ or ψ $ ϕ₂)
  (Prop_disj_right Γ ϕ₁ ϕ₂ ψ wfϕ₁ wfϕ₂ wfψ)
  BasicReasoning
  {
Σ: Signature
Γ: Theory
ϕ₁, ϕ₂, ψ: Pattern
wfϕ₁: well_formed ϕ₁
wfϕ₂: well_formed ϕ₂
wfψ: well_formed ψ
ProofInfoMeaning Γ
  (ψ $ (ϕ₁ or ϕ₂) ---> ψ $ ϕ₁ or ψ $ ϕ₂)
  (Prop_disj_right Γ ϕ₁ ϕ₂ ψ wfϕ₁ wfϕ₂ wfψ)
  BasicReasoning
    abstract (
      constructor; simpl;
      [(set_solver)|(set_solver)|(reflexivity)|(set_solver)]
    ).
  }
Defined.

Lemma prf_prop_or_iff {Σ : Signature} Γ AC p q:
  well_formed p ->
  well_formed q ->
  Γ ⊢i ((subst_ctx AC (p or q)) <---> ((subst_ctx AC p) or (subst_ctx AC q)))
  using (
  (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)).
Σ: Signature
Γ: Theory
AC: Application_context
p, q: Pattern
well_formed p
→ well_formed q
  → Γ ⊢i AC.[□↦p or q] <---> AC.[□↦p] or AC.[□↦q]
    using (ExGen := ∅, SVSubst := ∅, KT := false,
           FP := frames_of_AC AC)
Proof.
Σ: Signature
Γ: Theory
AC: Application_context
p, q: Pattern
well_formed p
→ well_formed q
  → Γ ⊢i AC.[□↦p or q] <---> AC.[□↦p] or AC.[□↦q]
    using (ExGen := ∅, SVSubst := ∅, KT := false,
           FP := frames_of_AC AC)
  intros wfp wfq.
Σ: Signature
Γ: Theory
AC: Application_context
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
Γ ⊢i AC.[□↦p or q] <---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
  induction AC; simpl.
Σ: Signature
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
Γ ⊢i p or q <---> p or q
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := ∅)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IHAC: Γ ⊢i AC.[□↦p or q] <---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i AC.[□↦p or q] $ p0 <--->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IHAC: Γ ⊢i AC.[□↦p or q] <---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦p or q] <--->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
  -
Σ: Signature
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
Γ ⊢i p or q <---> p or q
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := ∅) useBasicReasoning.
Σ: Signature
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
Γ ⊢i p or q <---> p or q using BasicReasoning apply pf_iff_equiv_refl; wf_auto2.
  -
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IHAC: Γ ⊢i AC.[□↦p or q] <---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i AC.[□↦p or q] $ p0 <--->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) apply pf_iff_iff in IHAC; try_wfauto2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IHAC: ((Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
  using (ExGen := ∅, SVSubst := ∅, KT := false,
         FP := frames_of_AC AC)) *
 (Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
  using (ExGen := ∅, SVSubst := ∅, KT := false,
         FP := frames_of_AC AC)))%type
Γ
⊢i AC.[□↦p or q] $ p0 <--->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    destruct IHAC as [IH1 IH2].
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ
⊢i AC.[□↦p or q] $ p0 <--->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    remember_constraint as i.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i AC.[□↦p or q] $ p0 <--->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i
    apply pf_iff_split; try_wfauto2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i AC.[□↦p or q] $ p0 --->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i AC.[□↦p] $ p0 or AC.[□↦q] $ p0 --->
   AC.[□↦p or q] $ p0 using i
    +
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i AC.[□↦p or q] $ p0 --->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i pose proof (H := IH1).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i AC.[□↦p or q] $ p0 --->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i
      apply useGenericReasoning with (i := i) in H.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using i
Γ
⊢i AC.[□↦p or q] $ p0 --->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC AC) i
      2: {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC AC) i subst i.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC AC)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) 
        apply pile_evs_svs_kt; auto.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
frames_of_AC AC ⊆ {[p0 ↾ Prf]} ∪ frames_of_AC AC set_solver.
      }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using i
Γ
⊢i AC.[□↦p or q] $ p0 --->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i
      rewrite Heqi in H.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i AC.[□↦p or q] $ p0 --->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i
      apply Framing_left with (ψ := p0) (wfψ := Prf) in H.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i AC.[□↦p or q] $ p0 --->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      2: {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) apply pile_evs_svs_kt; auto.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
{[p0 ↾ Prf]} ⊆ {[p0 ↾ Prf]} ∪ frames_of_AC AC set_solver. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i AC.[□↦p or q] $ p0 --->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i
      eapply syllogism_meta.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed (AC.[□↦p or q] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed ?B
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed (AC.[□↦p] $ p0 or AC.[□↦q] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i AC.[□↦p or q] $ p0 ---> ?B using i
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i ?B ---> AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i 4: subst i; apply H.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed (AC.[□↦p or q] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed ((AC.[□↦p] or AC.[□↦q]) $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed (AC.[□↦p] $ p0 or AC.[□↦q] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i (AC.[□↦p] or AC.[□↦q]) $ p0 --->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i
      all: try_wfauto2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i (AC.[□↦p] or AC.[□↦q]) $ p0 --->
   AC.[□↦p] $ p0 or AC.[□↦q] $ p0 using i
      remember (subst_ctx AC p) as p'.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] $ p0 ---> (p' or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i (p' or AC.[□↦q]) $ p0 ---> p' $ p0 or AC.[□↦q] $ p0
using i
      remember (subst_ctx AC q) as q'.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] $ p0 ---> (p' or q') $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i (p' or q') $ p0 ---> p' $ p0 or q' $ p0 using i
      subst i.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] $ p0 ---> (p' or q') $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i (p' or q') $ p0 ---> p' $ p0 or q' $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      eapply useGenericReasoning.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] $ p0 ---> (p' or q') $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
ProofInfoLe
  (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
   FP := ?fp)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] $ p0 ---> (p' or q') $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i (p' or q') $ p0 ---> p' $ p0 or q' $ p0
using (ExGen := ?evs, SVSubst := 
       ?svs, KT := ?kt, FP := 
       ?fp)
      2: eapply Prop_disj_left.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] $ p0 ---> (p' or q') $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false, FP := ∅)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] $ p0 ---> (p' or q') $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed p'
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] $ p0 ---> (p' or q') $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed q'
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] $ p0 ---> (p' or q') $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed p0 all: subst; try_wfauto2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false, FP := ∅)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false, FP := ∅)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) apply pile_evs_svs_kt; auto.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
H: Γ
⊢i AC.[□↦p or q] $ p0 --->
   (AC.[□↦p] or AC.[□↦q]) $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
∅ ⊆ {[p0 ↾ Prf]} ∪ frames_of_AC AC set_solver. }
    +
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i AC.[□↦p] $ p0 or AC.[□↦q] $ p0 --->
   AC.[□↦p or q] $ p0 using i eapply prf_disj_elim_meta_meta; try_wfauto2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i AC.[□↦p] $ p0 ---> AC.[□↦p or q] $ p0 using i
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i AC.[□↦q] $ p0 ---> AC.[□↦p or q] $ p0 using i
      *
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i AC.[□↦p] $ p0 ---> AC.[□↦p or q] $ p0 using i subst i.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] $ p0 ---> AC.[□↦p or q] $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) 
        apply Framing_left with (wfψ := Prf).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) apply pile_evs_svs_kt; auto.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
{[p0 ↾ Prf]} ⊆ {[p0 ↾ Prf]} ∪ frames_of_AC AC set_solver. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        eapply prf_weaken_conclusion_meta_meta.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p]
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed ?B
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p or q]
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i ?B ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> ?B
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        4: {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i ?B ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) gapply IH2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC AC)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p]
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed (AC.[□↦p] or AC.[□↦q])
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p or q]
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        1-3: wf_auto2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        useBasicReasoning.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> AC.[□↦p] or AC.[□↦q]
using BasicReasoning
        apply disj_left_intro; wf_auto2.
      *
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i AC.[□↦q] $ p0 ---> AC.[□↦p or q] $ p0 using i subst i.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] $ p0 ---> AC.[□↦p or q] $ p0
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        apply Framing_left with (wfψ := Prf).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        eapply prf_weaken_conclusion_meta_meta.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦q]
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed ?B
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p or q]
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i ?B ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> ?B
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) 4: gapply IH2; try_solve_pile.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦q]
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed (AC.[□↦p] or AC.[□↦q])
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p or q]
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) all: try_wfauto2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        useBasicReasoning.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> AC.[□↦p] or AC.[□↦q]
using BasicReasoning
        apply disj_right_intro; wf_auto2.
  -
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IHAC: Γ ⊢i AC.[□↦p or q] <---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦p or q] <--->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) apply pf_iff_iff in IHAC; try_wfauto2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IHAC: ((Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
  using (ExGen := ∅, SVSubst := ∅, KT := false,
         FP := frames_of_AC AC)) *
 (Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
  using (ExGen := ∅, SVSubst := ∅, KT := false,
         FP := frames_of_AC AC)))%type
Γ
⊢i p0 $ AC.[□↦p or q] <--->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    destruct IHAC as [IH1 IH2].
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦p or q] <--->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    remember_constraint as i.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦p or q] <--->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q] using i
    apply pf_iff_split; try_wfauto2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q] using i
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦p] or p0 $ AC.[□↦q] --->
   p0 $ AC.[□↦p or q] using i
    +
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q] using i pose proof (H := IH1).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q] using i
      apply useGenericReasoning with (i := i) in H.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using i
Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q] using i
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC AC) i
      2: {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC AC) i subst i.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := frames_of_AC AC)
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using i
Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q] using i
      eapply Framing_right with (ψ := p0)(wfψ := Prf) in H.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q] using i
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]}) i
      eapply syllogism_meta.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
well_formed (p0 $ AC.[□↦p or q])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
well_formed ?B
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
well_formed (p0 $ AC.[□↦p] or p0 $ AC.[□↦q])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
Γ ⊢i p0 $ AC.[□↦p or q] ---> ?B using i
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
Γ ⊢i ?B ---> p0 $ AC.[□↦p] or p0 $ AC.[□↦q] using i
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]}) i 4: apply H.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
well_formed (p0 $ AC.[□↦p or q])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
well_formed (p0 $ (AC.[□↦p] or AC.[□↦q]))
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
well_formed (p0 $ AC.[□↦p] or p0 $ AC.[□↦q])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
Γ
⊢i p0 $ (AC.[□↦p] or AC.[□↦q]) --->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q] using i
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]}) i
      all: try_wfauto2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
Γ
⊢i p0 $ (AC.[□↦p] or AC.[□↦q]) --->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q] using i
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]}) i
      2: {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]}) i subst i.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
H: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ
⊢i p0 $ AC.[□↦p or q] --->
   p0 $ (AC.[□↦p] or AC.[□↦q]) using i
Γ
⊢i p0 $ (AC.[□↦p] or AC.[□↦q]) --->
   p0 $ AC.[□↦p] or p0 $ AC.[□↦q] using i
      remember (subst_ctx AC p) as p'.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i p0 $ AC.[□↦p or q] ---> p0 $ (p' or AC.[□↦q])
using i
Γ
⊢i p0 $ (p' or AC.[□↦q]) ---> p0 $ p' or p0 $ AC.[□↦q]
using i
      remember (subst_ctx AC q) as q'.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
H: Γ ⊢i p0 $ AC.[□↦p or q] ---> p0 $ (p' or q')
using i
Γ ⊢i p0 $ (p' or q') ---> p0 $ p' or p0 $ q' using i
      subst i; apply useBasicReasoning.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
H: Γ ⊢i p0 $ AC.[□↦p or q] ---> p0 $ (p' or q')
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i p0 $ (p' or q') ---> p0 $ p' or p0 $ q'
using BasicReasoning
      apply Prop_disj_right.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
H: Γ ⊢i p0 $ AC.[□↦p or q] ---> p0 $ (p' or q')
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed p'
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
H: Γ ⊢i p0 $ AC.[□↦p or q] ---> p0 $ (p' or q')
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed q'
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
p': Pattern
Heqp': p' = AC.[□↦p]
q': Pattern
Heqq': q' = AC.[□↦q]
IH1: Γ ⊢i AC.[□↦p or q] ---> p' or q'
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i p' or q' ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
H: Γ ⊢i p0 $ AC.[□↦p or q] ---> p0 $ (p' or q')
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
well_formed p0 all: subst; try_wfauto2.
    +
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦p] or p0 $ AC.[□↦q] --->
   p0 $ AC.[□↦p or q] using i eapply prf_disj_elim_meta_meta; try_wfauto2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i p0 $ AC.[□↦p] ---> p0 $ AC.[□↦p or q] using i
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i p0 $ AC.[□↦q] ---> p0 $ AC.[□↦p or q] using i
      *
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i p0 $ AC.[□↦p] ---> p0 $ AC.[□↦p or q] using i subst i.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i p0 $ AC.[□↦p] ---> p0 $ AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        apply Framing_right with (wfψ := Prf).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        eapply prf_weaken_conclusion_meta_meta.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p]
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed ?B
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p or q]
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i ?B ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> ?B
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        4: gapply IH2; try_solve_pile.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p]
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed (AC.[□↦p] or AC.[□↦q])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p or q]
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) all: try_wfauto2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        useBasicReasoning.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦p] ---> AC.[□↦p] or AC.[□↦q]
using BasicReasoning
        apply disj_left_intro; wf_auto2.
      *
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
i: ProofInfo
Heqi: i =
(ExGen := ∅, SVSubst := ∅, KT := false,
 FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Γ ⊢i p0 $ AC.[□↦q] ---> p0 $ AC.[□↦p or q] using i subst i.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i p0 $ AC.[□↦q] ---> p0 $ AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        apply Framing_right with (wfψ := Prf).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        eapply prf_weaken_conclusion_meta_meta.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦q]
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed ?B
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p or q]
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i ?B ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> ?B
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        4: gapply IH2; try_solve_pile.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦q]
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed (AC.[□↦p] or AC.[□↦q])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
well_formed AC.[□↦p or q]
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        all: try_wfauto2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
        useBasicReasoning.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
IH1: Γ ⊢i AC.[□↦p or q] ---> AC.[□↦p] or AC.[□↦q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i AC.[□↦p] or AC.[□↦q] ---> AC.[□↦p or q]
using (ExGen := ∅, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Γ ⊢i AC.[□↦q] ---> AC.[□↦p] or AC.[□↦q]
using BasicReasoning
        apply disj_right_intro; wf_auto2.
Defined.




Lemma Singleton_ctx {Σ : Signature} (Γ : Theory) (C1 C2 : Application_context) (ϕ : Pattern) (x : evar) :
  well_formed ϕ ->
  Γ ⊢i (! ((subst_ctx C1 (patt_free_evar x and ϕ)) and
             (subst_ctx C2 (patt_free_evar x and (! ϕ)))))
  using BasicReasoning.
Σ: Signature
Γ: Theory
C1, C2: Application_context
ϕ: Pattern
x: evar
well_formed ϕ
→ Γ
  ⊢i ! (C1.[□↦patt_free_evar x and ϕ]
        and C2.[□↦patt_free_evar x and ! ϕ])
  using BasicReasoning
Proof.
Σ: Signature
Γ: Theory
C1, C2: Application_context
ϕ: Pattern
x: evar
well_formed ϕ
→ Γ
  ⊢i ! (C1.[□↦patt_free_evar x and ϕ]
        and C2.[□↦patt_free_evar x and ! ϕ])
  using BasicReasoning
  intros Hwf.
Σ: Signature
Γ: Theory
C1, C2: Application_context
ϕ: Pattern
x: evar
Hwf: well_formed ϕ
Γ
⊢i ! (C1.[□↦patt_free_evar x and ϕ]
      and C2.[□↦patt_free_evar x and ! ϕ])
using BasicReasoning
  unshelve (eexists).
Σ: Signature
Γ: Theory
C1, C2: Application_context
ϕ: Pattern
x: evar
Hwf: well_formed ϕ
ML_proof_system Γ
  (! (C1.[□↦patt_free_evar x and ϕ]
      and C2.[□↦patt_free_evar x and ! ϕ]))
Σ: Signature
Γ: Theory
C1, C2: Application_context
ϕ: Pattern
x: evar
Hwf: well_formed ϕ
ProofInfoMeaning Γ
  (! (C1.[□↦patt_free_evar x and ϕ]
      and C2.[□↦patt_free_evar x and ! ϕ])) ?pf
  BasicReasoning
  {
Σ: Signature
Γ: Theory
C1, C2: Application_context
ϕ: Pattern
x: evar
Hwf: well_formed ϕ
ML_proof_system Γ
  (! (C1.[□↦patt_free_evar x and ϕ]
      and C2.[□↦patt_free_evar x and ! ϕ]))
    apply ProofSystem.Singleton_ctx.
Σ: Signature
Γ: Theory
C1, C2: Application_context
ϕ: Pattern
x: evar
Hwf: well_formed ϕ
well_formed ϕ apply Hwf.
  }
Σ: Signature
Γ: Theory
C1, C2: Application_context
ϕ: Pattern
x: evar
Hwf: well_formed ϕ
ProofInfoMeaning Γ
  (! (C1.[□↦patt_free_evar x and ϕ]
      and C2.[□↦patt_free_evar x and ! ϕ]))
  (Singleton_ctx Γ C1 C2 ϕ x Hwf) BasicReasoning
  {
Σ: Signature
Γ: Theory
C1, C2: Application_context
ϕ: Pattern
x: evar
Hwf: well_formed ϕ
ProofInfoMeaning Γ
  (! (C1.[□↦patt_free_evar x and ϕ]
      and C2.[□↦patt_free_evar x and ! ϕ]))
  (Singleton_ctx Γ C1 C2 ϕ x Hwf) BasicReasoning
    abstract (
      constructor; simpl;
      [( set_solver )
      |( set_solver )
      |( reflexivity )
      |( set_solver )
      ]
    ).
  }
Defined.

Lemma Existence {Σ : Signature} (Γ : Theory) :
  Γ ⊢i (ex , patt_bound_evar 0) using BasicReasoning.
Σ: Signature
Γ: Theory
Γ ⊢i ex , b0 using BasicReasoning
Proof.
Σ: Signature
Γ: Theory
Γ ⊢i ex , b0 using BasicReasoning
  unshelve (eexists).
Σ: Signature
Γ: Theory
ML_proof_system Γ (ex , b0)
Σ: Signature
Γ: Theory
ProofInfoMeaning Γ (ex , b0) ?pf BasicReasoning
  {
Σ: Signature
Γ: Theory
ML_proof_system Γ (ex , b0)
    apply ProofSystem.Existence.
  }
Σ: Signature
Γ: Theory
ProofInfoMeaning Γ (ex , b0) (Existence Γ)
  BasicReasoning
  {
Σ: Signature
Γ: Theory
ProofInfoMeaning Γ (ex , b0) (Existence Γ)
  BasicReasoning
    abstract (
      constructor; simpl;
      [( set_solver )
      |( set_solver )
      |( reflexivity )
      |( set_solver )
      ]
    ).
  }
Defined.

Lemma Prop_ex_left {Σ : Signature} (Γ : Theory) (ϕ ψ : Pattern) :
  well_formed (ex, ϕ) ->
  well_formed ψ ->
  Γ ⊢i (ex , ϕ) $ ψ ---> ex , ϕ $ ψ
  using BasicReasoning.
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
well_formed (ex , ϕ)
→ well_formed ψ
  → Γ ⊢i (ex , ϕ) $ ψ ---> (ex , ϕ $ ψ)
    using BasicReasoning
Proof.
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
well_formed (ex , ϕ)
→ well_formed ψ
  → Γ ⊢i (ex , ϕ) $ ψ ---> (ex , ϕ $ ψ)
    using BasicReasoning
  intros wfϕ wfψ.
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
Γ ⊢i (ex , ϕ) $ ψ ---> (ex , ϕ $ ψ)
using BasicReasoning
  unshelve (eexists).
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
ML_proof_system Γ ((ex , ϕ) $ ψ ---> (ex , ϕ $ ψ))
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
ProofInfoMeaning Γ ((ex , ϕ) $ ψ ---> (ex , ϕ $ ψ))
  ?pf BasicReasoning
  {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
ML_proof_system Γ ((ex , ϕ) $ ψ ---> (ex , ϕ $ ψ))
    apply ProofSystem.Prop_ex_left.
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
well_formed (ex , ϕ)
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
well_formed ψ
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
well_formed (ex , ϕ) exact wfϕ. }
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
well_formed ψ
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
well_formed ψ exact wfψ. }
  }
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
ProofInfoMeaning Γ ((ex , ϕ) $ ψ ---> (ex , ϕ $ ψ))
  (Prop_ex_left Γ ϕ ψ wfϕ wfψ) BasicReasoning
  {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
ProofInfoMeaning Γ ((ex , ϕ) $ ψ ---> (ex , ϕ $ ψ))
  (Prop_ex_left Γ ϕ ψ wfϕ wfψ) BasicReasoning
    constructor; simpl.
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
true
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅ set_solver. }
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
true
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅ set_solver. }
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
true
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
true reflexivity. }
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅ set_solver. }
  }
Defined.

Lemma Prop_ex_right {Σ : Signature} (Γ : Theory) (ϕ ψ : Pattern) :
  well_formed (ex, ϕ) ->
  well_formed ψ ->
  Γ ⊢i ψ $ (ex , ϕ) ---> ex , ψ $ ϕ
  using BasicReasoning.
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
well_formed (ex , ϕ)
→ well_formed ψ
  → Γ ⊢i ψ $ (ex , ϕ) ---> (ex , ψ $ ϕ)
    using BasicReasoning
Proof.
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
well_formed (ex , ϕ)
→ well_formed ψ
  → Γ ⊢i ψ $ (ex , ϕ) ---> (ex , ψ $ ϕ)
    using BasicReasoning
  intros wfϕ wfψ.
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
Γ ⊢i ψ $ (ex , ϕ) ---> (ex , ψ $ ϕ)
using BasicReasoning
  unshelve (eexists).
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
ML_proof_system Γ (ψ $ (ex , ϕ) ---> (ex , ψ $ ϕ))
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
ProofInfoMeaning Γ (ψ $ (ex , ϕ) ---> (ex , ψ $ ϕ))
  ?pf BasicReasoning
  {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
ML_proof_system Γ (ψ $ (ex , ϕ) ---> (ex , ψ $ ϕ))
    apply ProofSystem.Prop_ex_right.
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
well_formed (ex , ϕ)
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
well_formed ψ
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
well_formed (ex , ϕ) exact wfϕ. }
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
well_formed ψ
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
well_formed ψ exact wfψ. }
  }
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
ProofInfoMeaning Γ (ψ $ (ex , ϕ) ---> (ex , ψ $ ϕ))
  (Prop_ex_right Γ ϕ ψ wfϕ wfψ) BasicReasoning
  {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
ProofInfoMeaning Γ (ψ $ (ex , ϕ) ---> (ex , ψ $ ϕ))
  (Prop_ex_right Γ ϕ ψ wfϕ wfψ) BasicReasoning
    constructor; simpl.
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
true
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅ set_solver. }
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
true
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅ set_solver. }
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
true
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
true reflexivity. }
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅
    {
Σ: Signature
Γ: Theory
ϕ, ψ: Pattern
wfϕ: well_formed (ex , ϕ)
wfψ: well_formed ψ
gset_to_coGset ∅ ⊆ ∅ set_solver. }
  }
Defined.


Tactic Notation "change" "constraint" "in" ident(H) :=
  let i := fresh "i" in
  remember_constraint as i;
  eapply useGenericReasoning with (i := i) in H;
  subst i;
  [|(try_solve_pile)].
 

Lemma prf_prop_ex_iff {Σ : Signature} Γ AC p x:
  evar_is_fresh_in x (subst_ctx AC p) ->
  well_formed (patt_exists p) = true ->
  Γ ⊢i ((subst_ctx AC (patt_exists p)) <---> (exists_quantify x (subst_ctx AC (p^{evar: 0 ↦ x}))))
  using (
  {| pi_generalized_evars := {[x]};
     pi_substituted_svars := ∅;
     pi_uses_kt := false ;
     pi_framing_patterns := frames_of_AC AC
  |}).
Σ: Signature
Γ: Theory
AC: Application_context
p: Pattern
x: evar
evar_is_fresh_in x AC.[□↦p]
→ well_formed (ex , p) = true
  → Γ
    ⊢i AC.[□↦ex , p] <--->
       exists_quantify x AC.[□↦p^{evar:0↦x}]
    using (ExGen := {[x]}, SVSubst := ∅, KT := false,
           FP := frames_of_AC AC)
Proof.
Σ: Signature
Γ: Theory
AC: Application_context
p: Pattern
x: evar
evar_is_fresh_in x AC.[□↦p]
→ well_formed (ex , p) = true
  → Γ
    ⊢i AC.[□↦ex , p] <--->
       exists_quantify x AC.[□↦p^{evar:0↦x}]
    using (ExGen := {[x]}, SVSubst := ∅, KT := false,
           FP := frames_of_AC AC)
  intros Hx Hwf.
Σ: Signature
Γ: Theory
AC: Application_context
p: Pattern
x: evar
Hx: evar_is_fresh_in x AC.[□↦p]
Hwf: well_formed (ex , p) = true
Γ
⊢i AC.[□↦ex , p] <--->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)

  induction AC; simpl.
Σ: Signature
Γ: Theory
p: Pattern
x: evar
Hx: evar_is_fresh_in x □.[□↦p]
Hwf: well_formed (ex , p) = true
Γ ⊢i (ex , p) <---> exists_quantify x p^{evar:0↦x}
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := ∅)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
  -
Σ: Signature
Γ: Theory
p: Pattern
x: evar
Hx: evar_is_fresh_in x □.[□↦p]
Hwf: well_formed (ex , p) = true
Γ ⊢i (ex , p) <---> exists_quantify x p^{evar:0↦x}
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := ∅) simpl in Hx.
Σ: Signature
Γ: Theory
p: Pattern
x: evar
Hx: evar_is_fresh_in x p
Hwf: well_formed (ex , p) = true
Γ ⊢i (ex , p) <---> exists_quantify x p^{evar:0↦x}
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := ∅)
    unfold exists_quantify.
Σ: Signature
Γ: Theory
p: Pattern
x: evar
Hx: evar_is_fresh_in x p
Hwf: well_formed (ex , p) = true
Γ ⊢i (ex , p) <---> (ex , p^{evar:0↦x}^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := ∅)
    erewrite evar_quantify_evar_open; auto.
Σ: Signature
Γ: Theory
p: Pattern
x: evar
Hx: evar_is_fresh_in x p
Hwf: well_formed (ex , p) = true
Γ ⊢i (ex , p) <---> (ex , p)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := ∅)
Σ: Signature
Γ: Theory
p: Pattern
x: evar
Hx: evar_is_fresh_in x p
Hwf: well_formed (ex , p) = true
well_formed_closed_ex_aux p 1 2: now do 2 apply andb_true_iff in Hwf as [_ Hwf].
Σ: Signature
Γ: Theory
p: Pattern
x: evar
Hx: evar_is_fresh_in x p
Hwf: well_formed (ex , p) = true
Γ ⊢i (ex , p) <---> (ex , p)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := ∅)
    useBasicReasoning.
Σ: Signature
Γ: Theory
p: Pattern
x: evar
Hx: evar_is_fresh_in x p
Hwf: well_formed (ex , p) = true
Γ ⊢i (ex , p) <---> (ex , p) using BasicReasoning
    apply pf_iff_equiv_refl.
Σ: Signature
Γ: Theory
p: Pattern
x: evar
Hx: evar_is_fresh_in x p
Hwf: well_formed (ex , p) = true
well_formed (ex , p) exact Hwf.
  -
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    assert (Hwfex: well_formed (ex , subst_ctx AC p)).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
well_formed (ex , AC.[□↦p])
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
well_formed (ex , AC.[□↦p]) unfold well_formed.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
well_formed_positive (ex , AC.[□↦p]) &&
well_formed_closed (ex , AC.[□↦p]) simpl.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p])
      pose proof (Hwf' := Hwf).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p])
      unfold well_formed in Hwf.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed_positive (ex , p) && well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p]) simpl in Hwf.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed_positive p && well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p])
      apply andb_prop in Hwf.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed_positive p = true ∧ well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p]) destruct Hwf as [Hwfp Hwfc].
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p])
      apply (wp_sctx AC p) in Hwfp.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfp: well_formed_positive AC.[□↦p] = true
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p]) rewrite Hwfp.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfp: well_formed_positive AC.[□↦p] = true
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
true && well_formed_closed (ex , AC.[□↦p]) simpl.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfp: well_formed_positive AC.[□↦p] = true
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed (ex , AC.[□↦p]) clear Hwfp.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed (ex , AC.[□↦p])
      unfold well_formed_closed.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux (ex , AC.[□↦p]) 0 &&
well_formed_closed_ex_aux (ex , AC.[□↦p]) 0 unfold well_formed_closed in Hwfc.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfc: well_formed_closed_mu_aux (ex , p) 0 &&
well_formed_closed_ex_aux (ex , p) 0 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux (ex , AC.[□↦p]) 0 &&
well_formed_closed_ex_aux (ex , AC.[□↦p]) 0 simpl in Hwfc.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux (ex , AC.[□↦p]) 0 &&
well_formed_closed_ex_aux (ex , AC.[□↦p]) 0 simpl.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux AC.[□↦p] 0 &&
well_formed_closed_ex_aux AC.[□↦p] 1
      split_and!.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux AC.[□↦p] 0 = true
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_ex_aux AC.[□↦p] 1 = true
      +
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux AC.[□↦p] 0 = true apply wcmu_sctx.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux p 0 = true destruct_and!.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux p 0 = true assumption.
      +
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_ex_aux AC.[□↦p] 1 = true apply wcex_sctx.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_ex_aux p 1 = true destruct_and!.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_ex_aux p 1 = true assumption.
    }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    assert(Hxfr1: evar_is_fresh_in x (subst_ctx AC p)).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
evar_is_fresh_in x AC.[□↦p]
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
evar_is_fresh_in x AC.[□↦p] simpl in Hx.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
evar_is_fresh_in x AC.[□↦p]
      eapply evar_is_fresh_in_richer.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
free_evars AC.[□↦p] ⊆ free_evars ?ϕ₂
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
evar_is_fresh_in x ?ϕ₂
      2: {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
evar_is_fresh_in x ?ϕ₂ apply Hx. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
free_evars AC.[□↦p] ⊆ free_evars (AC.[□↦p] $ p0)
      solve_free_evars_inclusion 5.
    }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC $ₗ p0).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    simpl in Hx.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    pose proof (Hxfr1' := Hxfr1).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1, Hxfr1': evar_is_fresh_in x AC.[□↦p]
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    rewrite -> evar_is_fresh_in_subst_ctx in Hxfr1'.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfr1': evar_is_fresh_in x p ∧ x ∉ AC_free_evars AC
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    destruct Hxfr1' as [Hxfrp HxAC].
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    assert(Hwf': well_formed (exists_quantify x (subst_ctx AC (p^{evar: 0 ↦ x})))).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
well_formed (exists_quantify x AC.[□↦p^{evar:0↦x}])
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
well_formed (exists_quantify x AC.[□↦p^{evar:0↦x}])
      unfold exists_quantify.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
well_formed (ex , AC.[□↦p^{evar:0↦x}]^{{evar:x↦0}})
      clear -HxAC Hwf.
Σ: Signature
AC: Application_context
p: Pattern
x: evar
Hwf: well_formed (ex , p) = true
HxAC: x ∉ AC_free_evars AC
well_formed (ex , AC.[□↦p^{evar:0↦x}]^{{evar:x↦0}})
      apply wf_ex_eq_sctx_eo.
Σ: Signature
AC: Application_context
p: Pattern
x: evar
Hwf: well_formed (ex , p) = true
HxAC: x ∉ AC_free_evars AC
well_formed (ex , p) = true
      apply Hwf.
    }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    (* TODO automate this *)
    assert (Hwfeo: well_formed (p^{evar: 0 ↦ x})).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed p^{evar:0↦x}
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed p^{evar:0↦x}
      unfold well_formed.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p^{evar:0↦x} &&
well_formed_closed p^{evar:0↦x}
      unfold well_formed,well_formed_closed in Hwf.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: [&& well_formed_positive (ex , p), well_formed_closed_mu_aux (ex , p) 0
 & well_formed_closed_ex_aux (ex , p) 0] = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p^{evar:0↦x} &&
well_formed_closed p^{evar:0↦x} apply andb_prop in Hwf.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed_positive (ex , p) = true
∧ well_formed_closed_mu_aux (ex , p) 0 &&
well_formed_closed_ex_aux (ex , p) 0 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p^{evar:0↦x} &&
well_formed_closed p^{evar:0↦x} simpl in Hwf.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p^{evar:0↦x} &&
well_formed_closed p^{evar:0↦x}
      rewrite wfp_evar_open.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p = true
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 &&
  well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
true && well_formed_closed p^{evar:0↦x}
      {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p = true apply Hwf. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
true && well_formed_closed p^{evar:0↦x}
      unfold well_formed_closed.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
[&& true, well_formed_closed_mu_aux p^{evar:0↦x} 0
  & well_formed_closed_ex_aux p^{evar:0↦x} 0]
      destruct_and!.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
H: well_formed_positive p = true
H1: well_formed_closed_mu_aux p 0 = true
H2: well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
[&& true, well_formed_closed_mu_aux p^{evar:0↦x} 0
  & well_formed_closed_ex_aux p^{evar:0↦x} 0]
      split_and!; auto.
    }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)


    (* TODO automate this. The problem is that [well_formed_app] and others do not have [= true];
       that is why [auto] does not work. But [auto] is not suitable for this anyway.
       A better way would be to create some `simpl_well_formed` tuple, that might use the type class
       mechanism for extension...
     *)
    assert(Hwf'p0: well_formed (exists_quantify x (subst_ctx AC (p^{evar: 0 ↦ x}) $ p0))).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed
  (exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0))
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed
  (exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0))
      unfold exists_quantify.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed
  (ex , (AC.[□↦p^{evar:0↦x}] $ p0)^{{evar:x↦0}})
      apply wf_ex_evar_quantify.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed (AC.[□↦p^{evar:0↦x}] $ p0) = true
      apply well_formed_app.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed AC.[□↦p^{evar:0↦x}] = true
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed p0 = true
      2: {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed p0 = true apply Prf. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed AC.[□↦p^{evar:0↦x}] = true
      auto.
    }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    apply pf_iff_iff in IHAC; auto.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IHAC: ((Γ
  ⊢i AC.[□↦ex , p] --->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)) *
 (Γ
  ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
     AC.[□↦ex , p]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)))%type
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    destruct IHAC as [IH1 IH2].
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦ex , p] $ p0 <--->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    apply pf_iff_split; auto.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0) --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    +
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) pose proof (H := IH1).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      change constraint in IH1.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      apply Framing_left with (ψ := p0) (wfψ := Prf) in IH1.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] $ p0 ---> exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      2: {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] $ p0 ---> exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

      eapply syllogism_meta.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] $ p0 ---> exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed (AC.[□↦ex , p] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed ?B
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed
  (exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0))
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ ⊢i AC.[□↦ex , p] $ p0 ---> ?B
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i ?B --->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) 4: apply IH1.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] $ p0 ---> exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed (AC.[□↦ex , p] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed
  (exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0))
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      1-3: wf_auto2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] $ p0 ---> exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

      remember (subst_ctx AC (p^{evar: 0 ↦ x})) as p'.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i exists_quantify x p' $ p0 --->
   exists_quantify x (p' $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      unfold exists_quantify.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i (ex , p'^{{evar:x↦0}}) $ p0 --->
   (ex , (p' $ p0)^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      simpl.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i (ex , p'^{{evar:x↦0}}) $ p0 --->
   (ex , p'^{{evar:x↦0}} $ p0^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) rewrite [p0^{{evar: x ↦ 0}}]evar_quantify_fresh.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
evar_is_fresh_in x p0
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i (ex , p'^{{evar:x↦0}}) $ p0 --->
   (ex , p'^{{evar:x↦0}} $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
evar_is_fresh_in x p0 eapply evar_is_fresh_in_app_r.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
evar_is_fresh_in x (?ϕ₁ $ p0) apply Hx. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i (ex , p'^{{evar:x↦0}}) $ p0 --->
   (ex , p'^{{evar:x↦0}} $ p0)
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      useBasicReasoning.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i (ex , p'^{{evar:x↦0}}) $ p0 --->
   (ex , p'^{{evar:x↦0}} $ p0) using BasicReasoning
      apply Prop_ex_left.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed (ex , p'^{{evar:x↦0}})
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed p0 wf_auto2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i AC.[□↦ex , p] $ p0 --->
   exists_quantify x p' $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p' $ p0))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed p0 wf_auto2.
    +
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0) --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) clear IH1.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0) --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

      change constraint in IH2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0) --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      apply Framing_left with (ψ := p0) (wfψ := Prf) in IH2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0) --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      2: {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0) --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      eapply syllogism_meta.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
well_formed
  (exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0))
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
well_formed ?B
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
well_formed (AC.[□↦ex , p] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0) --->
   ?B
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ ⊢i ?B ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) 5: eapply IH2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
well_formed
  (exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0))
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
well_formed (AC.[□↦ex , p] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0) --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      1-3: wf_auto2.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i exists_quantify x (AC.[□↦p^{evar:0↦x}] $ p0) --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

      apply Ex_gen; auto.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
ProofInfoLe
  (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := ∅)
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
x
∉ free_evars
    (exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p^{evar:0↦x}] $ p0 --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
ProofInfoLe
  (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := ∅)
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
x
∉ free_evars
    (exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p^{evar:0↦x}] $ p0 --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      1: {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
x
∉ free_evars
    (exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0)
        unfold exists_quantify.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
x
∉ free_evars
    ((ex , AC.[□↦p^{evar:0↦x}]^{{evar:x↦0}}) $ p0)
        simpl.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
x
∉ free_evars AC.[□↦p^{evar:0↦x}]^{{evar:x↦0}}
  ∪ free_evars p0
        rewrite free_evars_evar_quantify.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
x
∉ free_evars AC.[□↦p^{evar:0↦x}] ∖ {[x]}
  ∪ free_evars p0
        unfold evar_is_fresh_in in Hx.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: x ∉ free_evars (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
x
∉ free_evars AC.[□↦p^{evar:0↦x}] ∖ {[x]}
  ∪ free_evars p0 simpl in Hx.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: x ∉ free_evars AC.[□↦p] ∪ free_evars p0
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
x
∉ free_evars AC.[□↦p^{evar:0↦x}] ∖ {[x]}
  ∪ free_evars p0 clear -Hx.
Σ: Signature
AC: Application_context
p0, p: Pattern
x: evar
Hx: x ∉ free_evars AC.[□↦p] ∪ free_evars p0
x
∉ free_evars AC.[□↦p^{evar:0↦x}] ∖ {[x]}
  ∪ free_evars p0
        set_solver.
      }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p^{evar:0↦x}] $ p0 --->
   exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

      (* TODO have some nice implicit parameters *)
      gapply (Framing_left _ _ _ _ _ Prf).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
ProofInfoLe
  (ExGen := ?evs, SVSubst := ?svs, KT := ?kt,
   FP := ?fp)
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p^{evar:0↦x}] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := ?evs, SVSubst := 
       ?svs, KT := ?kt, FP := 
       ?fp)
      apply pile_refl.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p^{evar:0↦x}] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      Unshelve.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p^{evar:0↦x}] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) 2: {
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p^{evar:0↦x}] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      unfold evar_open.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p^[evar:0↦patt_free_evar x]] --->
   exists_quantify x
     AC.[□↦p^[evar:0↦patt_free_evar x]]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      rewrite subst_ctx_bevar_subst.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p]^[evar:0↦patt_free_evar x] --->
   exists_quantify x
     AC.[□↦p]^[evar:0↦patt_free_evar x]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      unfold exists_quantify.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p]^[evar:0↦patt_free_evar x] --->
   (ex ,
    AC.[□↦p]^[evar:0↦patt_free_evar x]^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) simpl.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p]^[evar:0↦patt_free_evar x] --->
   (ex ,
    AC.[□↦p]^[evar:0↦patt_free_evar x]^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      fold ((subst_ctx AC p)^{evar: 0 ↦ x}).
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ
⊢i AC.[□↦p]^{evar:0↦x} --->
   (ex , AC.[□↦p]^{evar:0↦x}^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      rewrite -> evar_quantify_evar_open; auto.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ ⊢i AC.[□↦p]^{evar:0↦x} ---> (ex , AC.[□↦p])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 --->
   AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
well_formed_closed_ex_aux AC.[□↦p] 1
      2: now do 2 apply andb_true_iff in Hwfex as [_ Hwfex].
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ ⊢i AC.[□↦p]^{evar:0↦x} ---> (ex , AC.[□↦p])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      useBasicReasoning.
Σ: Signature
Γ: Theory
AC: Application_context
p0: Pattern
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (AC.[□↦p] $ p0)
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] $ p0 ---> AC.[□↦ex , p] $ p0
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (AC.[□↦p^{evar:0↦x}] $ p0))
Γ ⊢i AC.[□↦p]^{evar:0↦x} ---> (ex , AC.[□↦p])
using BasicReasoning
      apply Ex_quan; auto.
  -
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    assert (Hwfex: well_formed (ex , subst_ctx AC p)).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
well_formed (ex , AC.[□↦p])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
well_formed (ex , AC.[□↦p]) unfold well_formed.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
well_formed_positive (ex , AC.[□↦p]) &&
well_formed_closed (ex , AC.[□↦p]) simpl.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p])
      pose proof (Hwf' := Hwf).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p])
      unfold well_formed in Hwf.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed_positive (ex , p) && well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p]) simpl in Hwf.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed_positive p && well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p])
      apply andb_prop in Hwf.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed_positive p = true ∧ well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p]) destruct Hwf as [Hwfp Hwfc].
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p])
      apply (wp_sctx AC p) in Hwfp.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfp: well_formed_positive AC.[□↦p] = true
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_positive AC.[□↦p] &&
well_formed_closed (ex , AC.[□↦p]) rewrite Hwfp.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfp: well_formed_positive AC.[□↦p] = true
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
true && well_formed_closed (ex , AC.[□↦p]) simpl.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfp: well_formed_positive AC.[□↦p] = true
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed (ex , AC.[□↦p]) clear Hwfp.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed (ex , AC.[□↦p])
      unfold well_formed_closed.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfc: well_formed_closed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux (ex , AC.[□↦p]) 0 &&
well_formed_closed_ex_aux (ex , AC.[□↦p]) 0 unfold well_formed_closed in Hwfc.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfc: well_formed_closed_mu_aux (ex , p) 0 &&
well_formed_closed_ex_aux (ex , p) 0 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux (ex , AC.[□↦p]) 0 &&
well_formed_closed_ex_aux (ex , AC.[□↦p]) 0 simpl in Hwfc.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux (ex , AC.[□↦p]) 0 &&
well_formed_closed_ex_aux (ex , AC.[□↦p]) 0 simpl.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux AC.[□↦p] 0 &&
well_formed_closed_ex_aux AC.[□↦p] 1
      split_and!.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux AC.[□↦p] 0 = true
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_ex_aux AC.[□↦p] 1 = true
      +
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux AC.[□↦p] 0 = true apply wcmu_sctx.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux p 0 = true destruct_and!.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_mu_aux p 0 = true assumption.
      +
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_ex_aux AC.[□↦p] 1 = true apply wcex_sctx.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_ex_aux p 1 = true destruct_and!.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwf': well_formed (ex , p) = true
well_formed_closed_ex_aux p 1 = true assumption.
    }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    assert(Hxfr1: evar_is_fresh_in x (subst_ctx AC p)).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
evar_is_fresh_in x AC.[□↦p]
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
evar_is_fresh_in x AC.[□↦p] simpl in Hx.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
evar_is_fresh_in x AC.[□↦p]
      eapply evar_is_fresh_in_richer.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
free_evars AC.[□↦p] ⊆ free_evars ?ϕ₂
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
evar_is_fresh_in x ?ϕ₂
      2: {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
evar_is_fresh_in x ?ϕ₂ apply Hx. }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
free_evars AC.[□↦p] ⊆ free_evars (p0 $ AC.[□↦p])
      solve_free_evars_inclusion 5.
    }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ᵣ AC).[□↦p]
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    simpl in Hx.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    pose proof (Hxfr1' := Hxfr1).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1, Hxfr1': evar_is_fresh_in x AC.[□↦p]
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    rewrite -> evar_is_fresh_in_subst_ctx in Hxfr1'.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfr1': evar_is_fresh_in x p ∧ x ∉ AC_free_evars AC
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    destruct Hxfr1' as [Hxfrp HxAC].
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    assert(Hwf': well_formed (exists_quantify x (subst_ctx AC (p^{evar: 0 ↦ x})))).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
well_formed (exists_quantify x AC.[□↦p^{evar:0↦x}])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
well_formed (exists_quantify x AC.[□↦p^{evar:0↦x}])
      unfold exists_quantify.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
well_formed (ex , AC.[□↦p^{evar:0↦x}]^{{evar:x↦0}})
      clear -HxAC Hwf.
Σ: Signature
AC: Application_context
p: Pattern
x: evar
Hwf: well_formed (ex , p) = true
HxAC: x ∉ AC_free_evars AC
well_formed (ex , AC.[□↦p^{evar:0↦x}]^{{evar:x↦0}})
      apply wf_ex_eq_sctx_eo.
Σ: Signature
AC: Application_context
p: Pattern
x: evar
Hwf: well_formed (ex , p) = true
HxAC: x ∉ AC_free_evars AC
well_formed (ex , p) = true
      apply Hwf.
    }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    (* TODO automate this *)
    assert (Hwfeo: well_formed (p^{evar: 0 ↦ x})).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed p^{evar:0↦x}
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed p^{evar:0↦x}
      unfold well_formed.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p^{evar:0↦x} &&
well_formed_closed p^{evar:0↦x}
      unfold well_formed,well_formed_closed in Hwf.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: [&& well_formed_positive (ex , p), well_formed_closed_mu_aux (ex , p) 0
 & well_formed_closed_ex_aux (ex , p) 0] = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p^{evar:0↦x} &&
well_formed_closed p^{evar:0↦x} apply andb_prop in Hwf.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed_positive (ex , p) = true
∧ well_formed_closed_mu_aux (ex , p) 0 &&
well_formed_closed_ex_aux (ex , p) 0 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p^{evar:0↦x} &&
well_formed_closed p^{evar:0↦x} simpl in Hwf.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p^{evar:0↦x} &&
well_formed_closed p^{evar:0↦x}
      rewrite wfp_evar_open.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p = true
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 &&
  well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
true && well_formed_closed p^{evar:0↦x}
      {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
well_formed_positive p = true apply Hwf. }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
true && well_formed_closed p^{evar:0↦x}
      unfold well_formed_closed.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
[&& true, well_formed_closed_mu_aux p^{evar:0↦x} 0
  & well_formed_closed_ex_aux p^{evar:0↦x} 0]
      destruct_and!.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
H: well_formed_positive p = true
H1: well_formed_closed_mu_aux p 0 = true
H2: well_formed_closed_ex_aux p 1 = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
[&& true, well_formed_closed_mu_aux p^{evar:0↦x} 0
  & well_formed_closed_ex_aux p^{evar:0↦x} 0]
      split_and!; auto.
    }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    (* TODO automate this. The problem is that [well_formed_app] and others do not have [= true];
       that is why [auto] does not work. But [auto] is not suitable for this anyway.
       A better way would be to create some `simpl_well_formed` tuple, that might use the type class
       mechanism for extension...
     *)
    assert(Hwf'p0: well_formed (exists_quantify x (p0 $ subst_ctx AC (p^{evar: 0 ↦ x})))).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed
  (exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]))
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed
  (exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]))
      unfold exists_quantify.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed
  (ex , (p0 $ AC.[□↦p^{evar:0↦x}])^{{evar:x↦0}})
      apply wf_ex_evar_quantify.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
well_formed (p0 $ AC.[□↦p^{evar:0↦x}]) = true
      apply well_formed_app; auto.
    }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: evar_is_fresh_in x AC.[□↦p]
→ Γ
  ⊢i AC.[□↦ex , p] <--->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    apply pf_iff_iff in IHAC; auto.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IHAC: ((Γ
  ⊢i AC.[□↦ex , p] --->
     exists_quantify x AC.[□↦p^{evar:0↦x}]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)) *
 (Γ
  ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
     AC.[□↦ex , p]
  using (ExGen := {[x]}, SVSubst := ∅,
         KT := false, FP := 
         frames_of_AC AC)))%type
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

    destruct IHAC as [IH1 IH2].
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i p0 $ AC.[□↦ex , p] <--->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    apply pf_iff_split; auto.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i p0 $ AC.[□↦ex , p] --->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]) --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
    +
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i p0 $ AC.[□↦ex , p] --->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) pose proof (H := IH1).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦ex , p] --->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      change constraint in IH1.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦ex , p] --->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      apply Framing_right with (wfψ := Prf) in IH1.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i p0 $ AC.[□↦ex , p] ---> p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦ex , p] --->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      2: try_solve_pile.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i p0 $ AC.[□↦ex , p] ---> p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ AC.[□↦ex , p] --->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      eapply syllogism_meta.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i p0 $ AC.[□↦ex , p] ---> p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed (p0 $ AC.[□↦ex , p])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed ?B
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed
  (exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]))
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ ⊢i p0 $ AC.[□↦ex , p] ---> ?B
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i ?B --->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) 4: apply IH1.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i p0 $ AC.[□↦ex , p] ---> p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed (p0 $ AC.[□↦ex , p])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed
  (p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed
  (exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]))
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      1-3: wf_auto2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i p0 $ AC.[□↦ex , p] ---> p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
H: Γ
⊢i AC.[□↦ex , p] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      remember (subst_ctx AC (p^{evar: 0 ↦ x})) as p'.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ exists_quantify x p' --->
   exists_quantify x (p0 $ p')
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      unfold exists_quantify.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ (ex , p'^{{evar:x↦0}}) --->
   (ex , (p0 $ p')^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      simpl.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ (ex , p'^{{evar:x↦0}}) --->
   (ex , p0^{{evar:x↦0}} $ p'^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) rewrite [p0^{{evar: x ↦ 0}}]evar_quantify_fresh.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
evar_is_fresh_in x p0
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ (ex , p'^{{evar:x↦0}}) --->
   (ex , p0 $ p'^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
evar_is_fresh_in x p0 eapply evar_is_fresh_in_app_l.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
evar_is_fresh_in x (p0 $ ?ϕ₂) apply Hx. }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ (ex , p'^{{evar:x↦0}}) --->
   (ex , p0 $ p'^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      useBasicReasoning.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
Γ
⊢i p0 $ (ex , p'^{{evar:x↦0}}) --->
   (ex , p0 $ p'^{{evar:x↦0}}) using BasicReasoning
      apply Prop_ex_right.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed (ex , p'^{{evar:x↦0}})
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed p0 wf_auto2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
p': Pattern
Heqp': p' = AC.[□↦p^{evar:0↦x}]
IH1: Γ
⊢i p0 $ AC.[□↦ex , p] --->
   p0 $ exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x p' ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false, FP := 
       frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed (exists_quantify x p')
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed (exists_quantify x (p0 $ p'))
H: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x p'
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := frames_of_AC AC)
well_formed p0 wf_auto2.
    +
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH1: Γ ⊢i AC.[□↦ex , p] ---> exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]) --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) clear IH1.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]) --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

      change constraint in IH2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]) --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      eapply (Framing_right _ _ _ _ _ Prf) in IH2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]) --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      eapply syllogism_meta.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
well_formed
  (exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]))
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
well_formed ?B
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
well_formed (p0 $ AC.[□↦ex , p])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]) --->
   ?B
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ ⊢i ?B ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) 5: eapply IH2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
well_formed
  (exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]))
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
well_formed
  (p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
well_formed (p0 $ AC.[□↦ex , p])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]) --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      1-3: wf_auto2.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]) --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      Unshelve.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]) --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      2: {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i exists_quantify x AC.[□↦p^{evar:0↦x}] ---> AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i exists_quantify x (p0 $ AC.[□↦p^{evar:0↦x}]) --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

      apply Ex_gen; auto.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
ProofInfoLe
  (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := ∅)
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
x
∉ free_evars
    (p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i p0 $ AC.[□↦p^{evar:0↦x}] --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
ProofInfoLe
  (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := ∅)
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
x
∉ free_evars
    (p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}])
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i p0 $ AC.[□↦p^{evar:0↦x}] --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      1: {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
x
∉ free_evars
    (p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}])
        unfold exists_quantify.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
x
∉ free_evars
    (p0 $ (ex , AC.[□↦p^{evar:0↦x}]^{{evar:x↦0}}))
        simpl.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
x
∉ free_evars p0
  ∪ free_evars AC.[□↦p^{evar:0↦x}]^{{evar:x↦0}}
        rewrite free_evars_evar_quantify.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
x
∉ free_evars p0
  ∪ free_evars AC.[□↦p^{evar:0↦x}] ∖ {[x]}
        unfold evar_is_fresh_in in Hx.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: x ∉ free_evars (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
x
∉ free_evars p0
  ∪ free_evars AC.[□↦p^{evar:0↦x}] ∖ {[x]} simpl in Hx.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: x ∉ free_evars p0 ∪ free_evars AC.[□↦p]
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
x
∉ free_evars p0
  ∪ free_evars AC.[□↦p^{evar:0↦x}] ∖ {[x]} clear -Hx.
Σ: Signature
p0: Pattern
AC: Application_context
p: Pattern
x: evar
Hx: x ∉ free_evars p0 ∪ free_evars AC.[□↦p]
x
∉ free_evars p0
  ∪ free_evars AC.[□↦p^{evar:0↦x}] ∖ {[x]}
        set_solver.
      }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i p0 $ AC.[□↦p^{evar:0↦x}] --->
   p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)

      eapply (Framing_right _ _ _ _ _ Prf).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i AC.[□↦p^{evar:0↦x}] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) Unshelve.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i AC.[□↦p^{evar:0↦x}] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      2: {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]})
  (ExGen := {[x]}, SVSubst := ∅, KT := false,
   FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) try_solve_pile. }
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i AC.[□↦p^{evar:0↦x}] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      {
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i AC.[□↦p^{evar:0↦x}] --->
   exists_quantify x AC.[□↦p^{evar:0↦x}]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      unfold evar_open.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i AC.[□↦p^[evar:0↦patt_free_evar x]] --->
   exists_quantify x
     AC.[□↦p^[evar:0↦patt_free_evar x]]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      rewrite subst_ctx_bevar_subst.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i AC.[□↦p]^[evar:0↦patt_free_evar x] --->
   exists_quantify x
     AC.[□↦p]^[evar:0↦patt_free_evar x]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      unfold exists_quantify.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i AC.[□↦p]^[evar:0↦patt_free_evar x] --->
   (ex ,
    AC.[□↦p]^[evar:0↦patt_free_evar x]^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC) simpl.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i AC.[□↦p]^[evar:0↦patt_free_evar x] --->
   (ex ,
    AC.[□↦p]^[evar:0↦patt_free_evar x]^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      fold ((subst_ctx AC p)^{evar: 0 ↦ x}).
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ
⊢i AC.[□↦p]^{evar:0↦x} --->
   (ex , AC.[□↦p]^{evar:0↦x}^{{evar:x↦0}})
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      erewrite evar_quantify_evar_open; auto.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ ⊢i AC.[□↦p]^{evar:0↦x} ---> (ex , AC.[□↦p])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ
⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] --->
   p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅,
       KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
well_formed_closed_ex_aux AC.[□↦p] 1
      2: now do 2 apply andb_true_iff in Hwfex as [_ Hwfex].
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ ⊢i AC.[□↦p]^{evar:0↦x} ---> (ex , AC.[□↦p])
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
       FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
      useBasicReasoning.
Σ: Signature
Γ: Theory
p0: Pattern
AC: Application_context
Prf: well_formed p0
p: Pattern
x: evar
Hx: evar_is_fresh_in x (p0 $ AC.[□↦p])
Hwf: well_formed (ex , p) = true
IH2: Γ ⊢i p0 $ exists_quantify x AC.[□↦p^{evar:0↦x}] ---> p0 $ AC.[□↦ex , p]
using (ExGen := {[x]}, SVSubst := ∅, KT := false,
FP := {[p0 ↾ Prf]} ∪ frames_of_AC AC)
Hwfex: well_formed (ex , AC.[□↦p])
Hxfr1: evar_is_fresh_in x AC.[□↦p]
Hxfrp: evar_is_fresh_in x p
HxAC: x ∉ AC_free_evars AC
Hwf': well_formed
  (exists_quantify x AC.[□↦p^{evar:0↦x}])
Hwfeo: well_formed p^{evar:0↦x}
Hwf'p0: well_formed
  (exists_quantify x
     (p0 $ AC.[□↦p^{evar:0↦x}]))
Γ ⊢i AC.[□↦p]^{evar:0↦x} ---> (ex , AC.[□↦p])
using BasicReasoning
      apply Ex_quan; auto.
      }
Defined.


Lemma pf_evar_open_free_evar_subst_equiv_sides {Σ : Signature} Γ x n ϕ p q E i:
  x <> E ->
  well_formed p = true ->
  well_formed q = true ->
  Γ ⊢i ϕ^{evar: n ↦ x}^[[evar: E ↦ p]] <---> ϕ^{evar: n ↦ x}^[[evar: E ↦ q]] using i ->
  Γ ⊢i ϕ^[[evar: E ↦ p]]^{evar: n ↦ x} <---> ϕ^[[evar: E ↦ q]]^{evar: n ↦ x} using i.
Σ: Signature
Γ: Theory
x: evar
n: db_index
ϕ, p, q: Pattern
E: evar
i: ProofInfo
x ≠ E
→ well_formed p = true
  → well_formed q = true
    → Γ
      ⊢i ϕ^{evar:n↦x}^[[evar:E↦p]] <--->
         ϕ^{evar:n↦x}^[[evar:E↦q]] using i
      → Γ
        ⊢i ϕ^[[evar:E↦p]]^{evar:n↦x} <--->
           ϕ^[[evar:E↦q]]^{evar:n↦x} using i
Proof.
Σ: Signature
Γ: Theory
x: evar
n: db_index
ϕ, p, q: Pattern
E: evar
i: ProofInfo
x ≠ E
→ well_formed p = true
  → well_formed q = true
    → Γ
      ⊢i ϕ^{evar:n↦x}^[[evar:E↦p]] <--->
         ϕ^{evar:n↦x}^[[evar:E↦q]] using i
      → Γ
        ⊢i ϕ^[[evar:E↦p]]^{evar:n↦x} <--->
           ϕ^[[evar:E↦q]]^{evar:n↦x} using i
  intros Hx wfp wfq H.
Σ: Signature
Γ: Theory
x: evar
n: db_index
ϕ, p, q: Pattern
E: evar
i: ProofInfo
Hx: x ≠ E
wfp: well_formed p = true
wfq: well_formed q = true
H: Γ
⊢i ϕ^{evar:n↦x}^[[evar:E↦p]] <--->
   ϕ^{evar:n↦x}^[[evar:E↦q]] using i
Γ
⊢i ϕ^[[evar:E↦p]]^{evar:n↦x} <--->
   ϕ^[[evar:E↦q]]^{evar:n↦x} using i
  unshelve (eapply (cast_proof' Γ _ _ _ _ H)).
Σ: Signature
Γ: Theory
x: evar
n: db_index
ϕ, p, q: Pattern
E: evar
i: ProofInfo
Hx: x ≠ E
wfp: well_formed p = true
wfq: well_formed q = true
H: Γ
⊢i ϕ^{evar:n↦x}^[[evar:E↦p]] <--->
   ϕ^{evar:n↦x}^[[evar:E↦q]] using i
(ϕ^[[evar:E↦p]]^{evar:n↦x} <--->
 ϕ^[[evar:E↦q]]^{evar:n↦x}) =
(ϕ^{evar:n↦x}^[[evar:E↦p]] <--->
 ϕ^{evar:n↦x}^[[evar:E↦q]])
  rewrite -> evar_open_free_evar_subst_swap by assumption.
Σ: Signature
Γ: Theory
x: evar
n: db_index
ϕ, p, q: Pattern
E: evar
i: ProofInfo
Hx: x ≠ E
wfp: well_formed p = true
wfq: well_formed q = true
H: Γ
⊢i ϕ^{evar:n↦x}^[[evar:E↦p]] <--->
   ϕ^{evar:n↦x}^[[evar:E↦q]] using i
(ϕ^{evar:n↦x}^[[evar:E↦p]] <--->
 ϕ^[[evar:E↦q]]^{evar:n↦x}) =
(ϕ^{evar:n↦x}^[[evar:E↦p]] <--->
 ϕ^{evar:n↦x}^[[evar:E↦q]])
  rewrite -> evar_open_free_evar_subst_swap by assumption.
Σ: Signature
Γ: Theory
x: evar
n: db_index
ϕ, p, q: Pattern
E: evar
i: ProofInfo
Hx: x ≠ E
wfp: well_formed p = true
wfq: well_formed q = true
H: Γ
⊢i ϕ^{evar:n↦x}^[[evar:E↦p]] <--->
   ϕ^{evar:n↦x}^[[evar:E↦q]] using i
(ϕ^{evar:n↦x}^[[evar:E↦p]] <--->
 ϕ^{evar:n↦x}^[[evar:E↦q]]) =
(ϕ^{evar:n↦x}^[[evar:E↦p]] <--->
 ϕ^{evar:n↦x}^[[evar:E↦q]])
  reflexivity.
Defined.

Lemma pf_iff_free_evar_subst_svar_open_to_bsvar_subst_free_evar_subst {Σ : Signature} Γ ϕ p q E X i:
  well_formed_closed_mu_aux p 0 = true ->
  well_formed_closed_mu_aux q 0 = true ->
  Γ ⊢i ϕ^{svar: 0 ↦ X}^[[evar: E ↦ p]] <---> ϕ^{svar: 0 ↦ X}^[[evar: E ↦ q]] using i ->
  Γ ⊢i ϕ^[[evar: E ↦ p]]^[svar: 0 ↦ patt_free_svar X] <--->
      ϕ^[[evar: E ↦ q]]^[svar: 0 ↦ patt_free_svar X] using i.
Σ: Signature
Γ: Theory
ϕ, p, q: Pattern
E: evar
X: svar
i: ProofInfo
well_formed_closed_mu_aux p 0 = true
→ well_formed_closed_mu_aux q 0 = true
  → Γ
    ⊢i ϕ^{svar:0↦X}^[[evar:E↦p]] <--->
       ϕ^{svar:0↦X}^[[evar:E↦q]] using i
    → Γ
      ⊢i ϕ^[[evar:E↦p]]^[svar:0↦patt_free_svar X] <--->
         ϕ^[[evar:E↦q]]^[svar:0↦patt_free_svar X]
      using i
Proof.
Σ: Signature
Γ: Theory
ϕ, p, q: Pattern
E: evar
X: svar
i: ProofInfo
well_formed_closed_mu_aux p 0 = true
→ well_formed_closed_mu_aux q 0 = true
  → Γ
    ⊢i ϕ^{svar:0↦X}^[[evar:E↦p]] <--->
       ϕ^{svar:0↦X}^[[evar:E↦q]] using i
    → Γ
      ⊢i ϕ^[[evar:E↦p]]^[svar:0↦patt_free_svar X] <--->
         ϕ^[[evar:E↦q]]^[svar:0↦patt_free_svar X]
      using i
  intros wfp wfq H.
Σ: Signature
Γ: Theory
ϕ, p, q: Pattern
E: evar
X: svar
i: ProofInfo
wfp: well_formed_closed_mu_aux p 0 = true
wfq: well_formed_closed_mu_aux q 0 = true
H: Γ
⊢i ϕ^{svar:0↦X}^[[evar:E↦p]] <--->
   ϕ^{svar:0↦X}^[[evar:E↦q]] using i
Γ
⊢i ϕ^[[evar:E↦p]]^[svar:0↦patt_free_svar X] <--->
   ϕ^[[evar:E↦q]]^[svar:0↦patt_free_svar X] using i
  unshelve (eapply (cast_proof' _ _ _ _ _ H)).
Σ: Signature
Γ: Theory
ϕ, p, q: Pattern
E: evar
X: svar
i: ProofInfo
wfp: well_formed_closed_mu_aux p 0 = true
wfq: well_formed_closed_mu_aux q 0 = true
H: Γ
⊢i ϕ^{svar:0↦X}^[[evar:E↦p]] <--->
   ϕ^{svar:0↦X}^[[evar:E↦q]] using i
(ϕ^[[evar:E↦p]]^[svar:0↦patt_free_svar X] <--->
 ϕ^[[evar:E↦q]]^[svar:0↦patt_free_svar X]) =
(ϕ^{svar:0↦X}^[[evar:E↦p]] <--->
 ϕ^{svar:0↦X}^[[evar:E↦q]])

  abstract (
    unfold svar_open in H;
    rewrite <- free_evar_subst_bsvar_subst;
    [idtac|wf_auto| unfold evar_is_fresh_in; simpl; clear; set_solver];
    rewrite <- free_evar_subst_bsvar_subst;
    [idtac|wf_auto|unfold evar_is_fresh_in; simpl; clear; set_solver];
    reflexivity
 ).
Defined.

Lemma pf_iff_mu_remove_svar_quantify_svar_open {Σ : Signature} Γ ϕ p q E X i :
  well_formed_closed_mu_aux (ϕ^[[evar: E ↦ p]]) 1 ->
  well_formed_closed_mu_aux (ϕ^[[evar: E ↦ q]]) 1 ->
  X ∉ free_svars (ϕ^[[evar: E ↦ p]]) ->
  X ∉ free_svars (ϕ^[[evar: E ↦ q]]) ->
  Γ ⊢i (mu , ϕ^[[evar: E ↦ p]]^{svar: 0 ↦ X}^{{svar: X ↦ 0}}) <--->
      (mu , ϕ^[[evar: E ↦ q]]^{svar: 0 ↦ X}^{{svar: X ↦ 0}}) using i ->
  Γ ⊢i (mu , ϕ^[[evar: E ↦ p]]) <---> (mu , ϕ^[[evar: E ↦ q]]) using i.
Σ: Signature
Γ: Theory
ϕ, p, q: Pattern
E: evar
X: svar
i: ProofInfo
well_formed_closed_mu_aux ϕ^[[evar:E↦p]] 1
→ well_formed_closed_mu_aux ϕ^[[evar:E↦q]] 1
  → X ∉ free_svars ϕ^[[evar:E↦p]]
    → X ∉ free_svars ϕ^[[evar:E↦q]]
      → Γ
        ⊢i (mu ,
            ϕ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}}) <--->
           (mu ,
            ϕ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
        using i
        → Γ
          ⊢i (mu , ϕ^[[evar:E↦p]]) <--->
             (mu , ϕ^[[evar:E↦q]]) using i
Proof.
Σ: Signature
Γ: Theory
ϕ, p, q: Pattern
E: evar
X: svar
i: ProofInfo
well_formed_closed_mu_aux ϕ^[[evar:E↦p]] 1
→ well_formed_closed_mu_aux ϕ^[[evar:E↦q]] 1
  → X ∉ free_svars ϕ^[[evar:E↦p]]
    → X ∉ free_svars ϕ^[[evar:E↦q]]
      → Γ
        ⊢i (mu ,
            ϕ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}}) <--->
           (mu ,
            ϕ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
        using i
        → Γ
          ⊢i (mu , ϕ^[[evar:E↦p]]) <--->
             (mu , ϕ^[[evar:E↦q]]) using i
  intros wfp' wfq' Xfrp Xfrq H.
Σ: Signature
Γ: Theory
ϕ, p, q: Pattern
E: evar
X: svar
i: ProofInfo
wfp': well_formed_closed_mu_aux ϕ^[[evar:E↦p]] 1
wfq': well_formed_closed_mu_aux ϕ^[[evar:E↦q]] 1
Xfrp: X ∉ free_svars ϕ^[[evar:E↦p]]
Xfrq: X ∉ free_svars ϕ^[[evar:E↦q]]
H: Γ
⊢i (mu , ϕ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}}) <--->
   (mu , ϕ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
using i
Γ ⊢i (mu , ϕ^[[evar:E↦p]]) <---> (mu , ϕ^[[evar:E↦q]])
using i
  unshelve (eapply (cast_proof' _ _ _ _ _ H)).
Σ: Signature
Γ: Theory
ϕ, p, q: Pattern
E: evar
X: svar
i: ProofInfo
wfp': well_formed_closed_mu_aux ϕ^[[evar:E↦p]] 1
wfq': well_formed_closed_mu_aux ϕ^[[evar:E↦q]] 1
Xfrp: X ∉ free_svars ϕ^[[evar:E↦p]]
Xfrq: X ∉ free_svars ϕ^[[evar:E↦q]]
H: Γ
⊢i (mu , ϕ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}}) <--->
   (mu , ϕ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
using i
((mu , ϕ^[[evar:E↦p]]) <---> (mu , ϕ^[[evar:E↦q]])) =
((mu , ϕ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}}) <--->
 (mu , ϕ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}}))
  abstract (
    rewrite -{1}[ϕ^[[evar: E ↦ p]]](@svar_quantify_svar_open _ X 0); [assumption| auto | auto];
    rewrite -{1}[ϕ^[[evar: E ↦ q]]](@svar_quantify_svar_open _ X 0); [assumption| auto | auto];
    reflexivity
  ).
Defined.


Add Search Blacklist "_elim".
Add Search Blacklist "_graph_rect".
Add Search Blacklist "_graph_mut".
Add Search Blacklist "FunctionalElimination_".


Section FOL_helpers.

  Context {Σ : Signature}.

  Fixpoint maximal_exists_depth_of_evar_in_pattern' (depth : nat) (E : evar) (ψ : Pattern) : nat :=
    match ψ with
    | patt_bott => 0
    | patt_sym _ => 0
    | patt_bound_evar _ => 0
    | patt_bound_svar _ => 0
    | patt_free_svar _ => 0
    | patt_free_evar E' =>
      match (decide (E' = E)) with
      | left _ => depth
      | right _ => 0
      end
    | patt_imp ψ₁ ψ₂
      => Nat.max
        (maximal_exists_depth_of_evar_in_pattern' depth E ψ₁)
        (maximal_exists_depth_of_evar_in_pattern' depth E ψ₂)
    | patt_app ψ₁ ψ₂
      => Nat.max
        (maximal_exists_depth_of_evar_in_pattern' depth E ψ₁)
        (maximal_exists_depth_of_evar_in_pattern' depth E ψ₂)
    | patt_exists ψ' =>
      maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ'
    | patt_mu ψ' =>
      maximal_exists_depth_of_evar_in_pattern' depth E ψ'
    end.

  Definition maximal_exists_depth_of_evar_in_pattern (E : evar) (ψ : Pattern) : nat :=
    maximal_exists_depth_of_evar_in_pattern' 0 E ψ.

  Definition pf_ite {P : Prop} (i : ProofInfo) (dec: {P} + {~P}) (Γ : Theory) (ϕ : Pattern)
    (pf1: P -> Γ ⊢i ϕ using i)
    (pf2: (~P) -> Γ ⊢i ϕ using i) :
    Γ ⊢i ϕ using i :=
    match dec with
    | left pf => pf1 pf
    | right pf => pf2 pf
    end.

  Lemma evar_fresh_seq_max (EvS : EVarSet) (n1 n2 : nat) :
    (@list_to_set evar EVarSet _ _ _ (evar_fresh_seq EvS n1)) ⊆ (list_to_set (evar_fresh_seq EvS (n1 `max` n2))).
Σ: Signature
EvS: EVarSet
n1, n2: nat
list_to_set (evar_fresh_seq EvS n1)
⊆ list_to_set (evar_fresh_seq EvS (n1 `max` n2))
  Proof.
Σ: Signature
EvS: EVarSet
n1, n2: nat
list_to_set (evar_fresh_seq EvS n1)
⊆ list_to_set (evar_fresh_seq EvS (n1 `max` n2))
    move: EvS n2.
Σ: Signature
n1: nat
∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set (evar_fresh_seq EvS (n1 `max` n2))
    induction n1; intros EvS n2.
Σ: Signature
EvS: EVarSet
n2: nat
list_to_set (evar_fresh_seq EvS 0)
⊆ list_to_set (evar_fresh_seq EvS (0 `max` n2))
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
list_to_set (evar_fresh_seq EvS (S n1))
⊆ list_to_set (evar_fresh_seq EvS (S n1 `max` n2))
    {
Σ: Signature
EvS: EVarSet
n2: nat
list_to_set (evar_fresh_seq EvS 0)
⊆ list_to_set (evar_fresh_seq EvS (0 `max` n2))
      simpl.
Σ: Signature
EvS: EVarSet
n2: nat
∅ ⊆ list_to_set (evar_fresh_seq EvS n2) set_solver.
    }
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
list_to_set (evar_fresh_seq EvS (S n1))
⊆ list_to_set (evar_fresh_seq EvS (S n1 `max` n2))
    {
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
list_to_set (evar_fresh_seq EvS (S n1))
⊆ list_to_set (evar_fresh_seq EvS (S n1 `max` n2))
      simpl.
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
{[evar_fresh_s EvS]}
∪ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ list_to_set (evar_fresh_seq EvS (S n1 `max` n2))
      destruct n2.
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
{[evar_fresh_s EvS]}
∪ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ list_to_set (evar_fresh_seq EvS (S n1 `max` 0))
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
{[evar_fresh_s EvS]}
∪ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ list_to_set (evar_fresh_seq EvS (S n1 `max` S n2))
      {
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
{[evar_fresh_s EvS]}
∪ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ list_to_set (evar_fresh_seq EvS (S n1 `max` 0))
        simpl.
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
{[evar_fresh_s EvS]}
∪ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ {[evar_fresh_s EvS]}
  ∪ list_to_set
      (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1) set_solver.
      }
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
{[evar_fresh_s EvS]}
∪ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ list_to_set (evar_fresh_seq EvS (S n1 `max` S n2))
      {
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
{[evar_fresh_s EvS]}
∪ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ list_to_set (evar_fresh_seq EvS (S n1 `max` S n2))
        simpl.
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
{[evar_fresh_s EvS]}
∪ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ {[evar_fresh_s EvS]}
  ∪ list_to_set
      (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS)
         (n1 `max` n2))
        cut (@list_to_set evar EVarSet _ _ _ (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
        ⊆ list_to_set (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) (n1 `max` n2))).
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
list_to_set
  (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS)
       (n1 `max` n2))
→ {[evar_fresh_s EvS]}
  ∪ list_to_set
      (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
  ⊆ {[evar_fresh_s EvS]}
    ∪ list_to_set
        (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS)
           (n1 `max` n2))
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
list_to_set
  (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS)
       (n1 `max` n2))
        {
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
list_to_set
  (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS)
       (n1 `max` n2))
→ {[evar_fresh_s EvS]}
  ∪ list_to_set
      (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
  ⊆ {[evar_fresh_s EvS]}
    ∪ list_to_set
        (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS)
           (n1 `max` n2))
          set_solver.
        }
Σ: Signature
n1: nat
IHn1: ∀ (EvS : EVarSet) (n2 : nat),
  list_to_set (evar_fresh_seq EvS n1)
  ⊆ list_to_set
      (evar_fresh_seq EvS (n1 `max` n2))
EvS: EVarSet
n2: nat
list_to_set
  (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS)
       (n1 `max` n2))
        specialize (IHn1 ({[evar_fresh_s EvS]} ∪ EvS) n2).
Σ: Signature
n1: nat
EvS: EVarSet
n2: nat
IHn1: list_to_set
  (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS)
     n1)
⊆ list_to_set
    (evar_fresh_seq
       ({[evar_fresh_s EvS]} ∪ EvS)
       (n1 `max` n2))
list_to_set
  (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS) n1)
⊆ list_to_set
    (evar_fresh_seq ({[evar_fresh_s EvS]} ∪ EvS)
       (n1 `max` n2))
        apply IHn1.
      }
    }
  Qed.

  Lemma medoeip_evar_open depth E n x ψ:
    E <> x ->
    maximal_exists_depth_of_evar_in_pattern' depth E (ψ^{evar: n ↦ x})
    = maximal_exists_depth_of_evar_in_pattern' depth E ψ.
Σ: Signature
depth: nat
E: evar
n: db_index
x: evar
ψ: Pattern
E ≠ x
→ maximal_exists_depth_of_evar_in_pattern' depth E
    ψ^{evar:n↦x} =
  maximal_exists_depth_of_evar_in_pattern' depth E ψ
  Proof.
Σ: Signature
depth: nat
E: evar
n: db_index
x: evar
ψ: Pattern
E ≠ x
→ maximal_exists_depth_of_evar_in_pattern' depth E
    ψ^{evar:n↦x} =
  maximal_exists_depth_of_evar_in_pattern' depth E ψ
    intros HEnex.
Σ: Signature
depth: nat
E: evar
n: db_index
x: evar
ψ: Pattern
HEnex: E ≠ x
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^{evar:n↦x} =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    unfold evar_open.
Σ: Signature
depth: nat
E: evar
n: db_index
x: evar
ψ: Pattern
HEnex: E ≠ x
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[evar:n↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    move: depth n.
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth E
    ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth E ψ
    induction ψ; intros depth n'; simpl; try reflexivity.
Σ: Signature
E, x: evar
n: db_index
HEnex: E ≠ x
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = 0
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
HEnex: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
HEnex: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ^[evar:S n'↦
  patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    {
Σ: Signature
E, x: evar
n: db_index
HEnex: E ≠ x
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = 0
      case_match; simpl; try reflexivity.
Σ: Signature
E, x: evar
n: db_index
HEnex: E ≠ x
depth: nat
n': db_index
e: n = n'
H: compare_nat n n' = Nat_equal n n' e
(if decide (x = E) then depth else 0) = 0
      case_match; try reflexivity.
Σ: Signature
E, x: evar
n: db_index
HEnex: E ≠ x
depth: nat
n': db_index
e: n = n'
H: compare_nat n n' = Nat_equal n n' e
e0: x = E
H0: decide (x = E) = left e0
depth = 0 subst.
Σ: Signature
E: evar
HEnex: E ≠ E
depth: nat
n': db_index
H: compare_nat n' n' = Nat_equal n' n' erefl
H0: decide (E = E) = left erefl
depth = 0 contradiction.
    }
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
HEnex: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
HEnex: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ^[evar:S n'↦
  patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    {
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
HEnex: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
      rewrite IHψ1.
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
HEnex: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 rewrite IHψ2.
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
HEnex: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 reflexivity.
    }
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
HEnex: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ^[evar:S n'↦
  patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    {
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
HEnex: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
      rewrite IHψ1.
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
HEnex: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 rewrite IHψ2.
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
HEnex: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 reflexivity.
    }
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ^[evar:S n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    {
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ^[evar:S n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ
      rewrite IHψ.
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ reflexivity.
    }
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    {
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[evar:n'↦patt_free_evar x] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
      rewrite IHψ.
Σ: Signature
E, x: evar
ψ: Pattern
HEnex: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[evar:n↦patt_free_evar x] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E ψ =
maximal_exists_depth_of_evar_in_pattern' depth E ψ reflexivity.
    }
  Qed.

  Lemma medoeip_svar_open depth E n X ψ:
    maximal_exists_depth_of_evar_in_pattern' depth E (ψ^{svar: n ↦ X})
    = maximal_exists_depth_of_evar_in_pattern' depth E ψ.
Σ: Signature
depth: nat
E: evar
n: db_index
X: svar
ψ: Pattern
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^{svar:n↦X} =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
  Proof.
Σ: Signature
depth: nat
E: evar
n: db_index
X: svar
ψ: Pattern
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^{svar:n↦X} =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    unfold svar_open.
Σ: Signature
depth: nat
E: evar
n: db_index
X: svar
ψ: Pattern
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[svar:n↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    move: depth n.
Σ: Signature
E: evar
X: svar
ψ: Pattern
∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth E
    ψ^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern' depth E ψ
    induction ψ; intros depth n'; simpl; try reflexivity.
Σ: Signature
E: evar
X: svar
n: db_index
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = 0
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ^[svar:n'↦
  patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[svar:S n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    {
Σ: Signature
E: evar
X: svar
n: db_index
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = 0
      case_match; simpl; try reflexivity.
    }
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ^[svar:n'↦
  patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[svar:S n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    {
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
      rewrite IHψ1.
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 rewrite IHψ2.
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 reflexivity.
    }
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ^[svar:n'↦
  patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[svar:S n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    {
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2
      rewrite IHψ1.
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 rewrite IHψ2.
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern'
    depth E ψ2
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 =
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 reflexivity.
    }
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_exists_depth_of_evar_in_pattern' depth E
ψ^[svar:n↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ^[svar:n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ^[svar:n↦patt_free_svar X] =
  maximal_exists_depth_of_evar_in_pattern' depth
    E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[svar:S n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    {
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_exists_depth_of_evar_in_pattern' depth E
ψ^[svar:n↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ^[svar:n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ
      rewrite IHψ.
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_exists_depth_of_evar_in_pattern' depth E
ψ^[svar:n↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ reflexivity.
    }
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_exists_depth_of_evar_in_pattern' depth E
ψ^[svar:n↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[svar:S n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
    {
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_exists_depth_of_evar_in_pattern' depth E
ψ^[svar:n↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E
  ψ^[svar:S n'↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
      rewrite IHψ.
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_exists_depth_of_evar_in_pattern' depth E
ψ^[svar:n↦patt_free_svar X] =
maximal_exists_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_exists_depth_of_evar_in_pattern' depth E ψ =
maximal_exists_depth_of_evar_in_pattern' depth E ψ reflexivity.
    }
  Qed.

  Lemma medoeip_notin E ψ depth:
    E ∉ free_evars ψ ->
    maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0.
Σ: Signature
E: evar
ψ: Pattern
depth: nat
E ∉ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' depth E ψ =
  0
  Proof.
Σ: Signature
E: evar
ψ: Pattern
depth: nat
E ∉ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' depth E ψ =
  0
    intros Hnotin.
Σ: Signature
E: evar
ψ: Pattern
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
    move: E depth Hnotin.
Σ: Signature
ψ: Pattern
∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern' depth E ψ =
    0
    induction ψ; intros E depth Hnotin; simpl in *; try reflexivity.
Σ: Signature
x, E: evar
depth: nat
Hnotin: E ∉ {[x]}
(if decide (x = E) then depth else 0) = 0
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 = 0
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 = 0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
    {
Σ: Signature
x, E: evar
depth: nat
Hnotin: E ∉ {[x]}
(if decide (x = E) then depth else 0) = 0 case_match.
Σ: Signature
x, E: evar
depth: nat
Hnotin: E ∉ {[x]}
e: x = E
H: decide (x = E) = left e
depth = 0
Σ: Signature
x, E: evar
depth: nat
Hnotin: E ∉ {[x]}
n: x ≠ E
H: decide (x = E) = right n
0 = 0 set_solver.
Σ: Signature
x, E: evar
depth: nat
Hnotin: E ∉ {[x]}
n: x ≠ E
H: decide (x = E) = right n
0 = 0 reflexivity. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 = 0
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 = 0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 = 0 rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 = 0 set_solver.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 = 0 rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0 `max` 0 = 0 set_solver.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0 `max` 0 = 0 reflexivity. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 = 0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 = 0 rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 = 0 set_solver.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0
`max` maximal_exists_depth_of_evar_in_pattern' depth E
        ψ2 = 0 rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0 `max` 0 = 0 set_solver.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0 `max` 0 = 0 reflexivity. }
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
    {
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
0 rewrite IHψ.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
E ∉ free_evars ψ
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
0 = 0 exact Hnotin.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
0 = 0 reflexivity. }
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
    {
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0 rewrite IHψ.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
E ∉ free_evars ψ
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
0 = 0 exact Hnotin.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_exists_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
0 = 0 reflexivity. }
  Qed.

  Lemma medoeip_S_in E ψ depth:
    E ∈ free_evars ψ ->
    maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ
    = S (maximal_exists_depth_of_evar_in_pattern' depth E ψ).
Σ: Signature
E: evar
ψ: Pattern
depth: nat
E ∈ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' (S depth) E
    ψ =
  S
    (maximal_exists_depth_of_evar_in_pattern' depth E
       ψ)
  Proof.
Σ: Signature
E: evar
ψ: Pattern
depth: nat
E ∈ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' (S depth) E
    ψ =
  S
    (maximal_exists_depth_of_evar_in_pattern' depth E
       ψ)
    intros Hin.
Σ: Signature
E: evar
ψ: Pattern
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
    move: E depth Hin.
Σ: Signature
ψ: Pattern
∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern' (S depth)
      E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern' depth
         E ψ)
    induction ψ; intros E depth Hin; simpl in *.
Σ: Signature
x, E: evar
depth: nat
Hin: E ∈ {[x]}
(if decide (x = E) then S depth else 0) =
S (if decide (x = E) then depth else 0)
Σ: Signature
x: svar
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' 
  (S (S depth)) E ψ =
S
  (maximal_exists_depth_of_evar_in_pattern' 
     (S depth) E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
    {
Σ: Signature
x, E: evar
depth: nat
Hin: E ∈ {[x]}
(if decide (x = E) then S depth else 0) =
S (if decide (x = E) then depth else 0) case_match.
Σ: Signature
x, E: evar
depth: nat
Hin: E ∈ {[x]}
e: x = E
H: decide (x = E) = left e
S depth = S depth
Σ: Signature
x, E: evar
depth: nat
Hin: E ∈ {[x]}
n: x ≠ E
H: decide (x = E) = right n
0 = 1 reflexivity.
Σ: Signature
x, E: evar
depth: nat
Hin: E ∈ {[x]}
n: x ≠ E
H: decide (x = E) = right n
0 = 1 set_solver. }
Σ: Signature
x: svar
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' 
  (S (S depth)) E ψ =
S
  (maximal_exists_depth_of_evar_in_pattern' 
     (S depth) E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
    {
Σ: Signature
x: svar
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1 set_solver. }
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' 
  (S (S depth)) E ψ =
S
  (maximal_exists_depth_of_evar_in_pattern' 
     (S depth) E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
    {
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1 set_solver. }
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' 
  (S (S depth)) E ψ =
S
  (maximal_exists_depth_of_evar_in_pattern' 
     (S depth) E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
    {
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1 set_solver. }
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' 
  (S (S depth)) E ψ =
S
  (maximal_exists_depth_of_evar_in_pattern' 
     (S depth) E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
    {
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1 set_solver. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' 
  (S (S depth)) E ψ =
S
  (maximal_exists_depth_of_evar_in_pattern' 
     (S depth) E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
      destruct (decide (E ∈ free_evars ψ1)),(decide (E ∈ free_evars ψ2)).
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
E ∈ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
E ∈ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) reflexivity.
      }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∈ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite [maximal_exists_depth_of_evar_in_pattern' _ _ ψ2]medoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite [maximal_exists_depth_of_evar_in_pattern' _ _ ψ2]medoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
        {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
        simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
        rewrite Nat.max_comm.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1) =
S
  (0
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ1)
        simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
        reflexivity.
      }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∈ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite [maximal_exists_depth_of_evar_in_pattern' _ _ ψ1]medoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite [maximal_exists_depth_of_evar_in_pattern' _ _ ψ1]medoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (0
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (0
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ2)
        reflexivity.
      }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        exfalso.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
False set_solver.
      }
    }
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' 
  (S (S depth)) E ψ =
S
  (maximal_exists_depth_of_evar_in_pattern' 
     (S depth) E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
    {
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1 set_solver. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' 
  (S (S depth)) E ψ =
S
  (maximal_exists_depth_of_evar_in_pattern' 
     (S depth) E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
      destruct (decide (E ∈ free_evars ψ1)),(decide (E ∈ free_evars ψ2)).
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
E ∈ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
E ∈ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) reflexivity.
      }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∈ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite [maximal_exists_depth_of_evar_in_pattern' _ _ ψ2]medoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite [maximal_exists_depth_of_evar_in_pattern' _ _ ψ2]medoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
        {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
        simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
        rewrite Nat.max_comm.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1) =
S
  (0
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ1)
        simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1)
        reflexivity.
      }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∈ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' 
  (S depth) E ψ1
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite [maximal_exists_depth_of_evar_in_pattern' _ _ ψ1]medoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        rewrite [maximal_exists_depth_of_evar_in_pattern' _ _ ψ1]medoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (0
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2) =
S
  (0
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ2) =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ2)
        reflexivity.
      }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_exists_depth_of_evar_in_pattern' (S depth) E
  ψ1
`max` maximal_exists_depth_of_evar_in_pattern'
        (S depth) E ψ2 =
S
  (maximal_exists_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_exists_depth_of_evar_in_pattern'
           depth E ψ2)
        exfalso.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
False set_solver.
      }     
    }
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S (S depth))
  E ψ =
S
  (maximal_exists_depth_of_evar_in_pattern' (S depth)
     E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
    {
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S (S depth))
  E ψ =
S
  (maximal_exists_depth_of_evar_in_pattern' (S depth)
     E ψ)
      rewrite IHψ.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
E ∈ free_evars ψ
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
S
  (maximal_exists_depth_of_evar_in_pattern' 
     (S depth) E ψ) =
S
  (maximal_exists_depth_of_evar_in_pattern' 
     (S depth) E ψ) assumption.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
S
  (maximal_exists_depth_of_evar_in_pattern' (S depth)
     E ψ) =
S
  (maximal_exists_depth_of_evar_in_pattern' (S depth)
     E ψ) reflexivity.
    }
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
    {
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
      rewrite IHψ.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
E ∈ free_evars ψ
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_exists_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_exists_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ) =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ) assumption.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_exists_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ) =
S (maximal_exists_depth_of_evar_in_pattern' depth E ψ) reflexivity.
    }
  Qed.


  Fixpoint maximal_mu_depth_of_evar_in_pattern' (depth : nat) (E : evar) (ψ : Pattern) : nat :=
    match ψ with
    | patt_bott => 0
    | patt_sym _ => 0
    | patt_bound_evar _ => 0
    | patt_bound_svar _ => 0
    | patt_free_svar _ => 0
    | patt_free_evar E' =>
      match (decide (E' = E)) with
      | left _ => depth
      | right _ => 0
      end
    | patt_imp ψ₁ ψ₂
      => Nat.max
        (maximal_mu_depth_of_evar_in_pattern' depth E ψ₁)
        (maximal_mu_depth_of_evar_in_pattern' depth E ψ₂)
    | patt_app ψ₁ ψ₂
      => Nat.max
        (maximal_mu_depth_of_evar_in_pattern' depth E ψ₁)
        (maximal_mu_depth_of_evar_in_pattern' depth E ψ₂)
    | patt_exists ψ' =>
      maximal_mu_depth_of_evar_in_pattern' depth E ψ'
    | patt_mu ψ' =>
      maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ'
    end.

  Definition maximal_mu_depth_of_evar_in_pattern (E : evar) (ψ : Pattern) : nat :=
    maximal_mu_depth_of_evar_in_pattern' 0 E ψ.



  Lemma svar_fresh_seq_max (SvS : SVarSet) (n1 n2 : nat) :
    (@list_to_set svar SVarSet _ _ _ (svar_fresh_seq SvS n1)) ⊆ (list_to_set (svar_fresh_seq SvS (n1 `max` n2))).
Σ: Signature
SvS: SVarSet
n1, n2: nat
list_to_set (svar_fresh_seq SvS n1)
⊆ list_to_set (svar_fresh_seq SvS (n1 `max` n2))
  Proof.
Σ: Signature
SvS: SVarSet
n1, n2: nat
list_to_set (svar_fresh_seq SvS n1)
⊆ list_to_set (svar_fresh_seq SvS (n1 `max` n2))
    move: SvS n2.
Σ: Signature
n1: nat
∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set (svar_fresh_seq SvS (n1 `max` n2))
    induction n1; intros SvS n2.
Σ: Signature
SvS: SVarSet
n2: nat
list_to_set (svar_fresh_seq SvS 0)
⊆ list_to_set (svar_fresh_seq SvS (0 `max` n2))
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
list_to_set (svar_fresh_seq SvS (S n1))
⊆ list_to_set (svar_fresh_seq SvS (S n1 `max` n2))
    {
Σ: Signature
SvS: SVarSet
n2: nat
list_to_set (svar_fresh_seq SvS 0)
⊆ list_to_set (svar_fresh_seq SvS (0 `max` n2))
      simpl.
Σ: Signature
SvS: SVarSet
n2: nat
∅ ⊆ list_to_set (svar_fresh_seq SvS n2) set_solver.
    }
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
list_to_set (svar_fresh_seq SvS (S n1))
⊆ list_to_set (svar_fresh_seq SvS (S n1 `max` n2))
    {
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
list_to_set (svar_fresh_seq SvS (S n1))
⊆ list_to_set (svar_fresh_seq SvS (S n1 `max` n2))
      simpl.
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
{[svar_fresh_s SvS]}
∪ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ list_to_set (svar_fresh_seq SvS (S n1 `max` n2))
      destruct n2.
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
{[svar_fresh_s SvS]}
∪ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ list_to_set (svar_fresh_seq SvS (S n1 `max` 0))
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
{[svar_fresh_s SvS]}
∪ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ list_to_set (svar_fresh_seq SvS (S n1 `max` S n2))
      {
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
{[svar_fresh_s SvS]}
∪ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ list_to_set (svar_fresh_seq SvS (S n1 `max` 0))
        simpl.
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
{[svar_fresh_s SvS]}
∪ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ {[svar_fresh_s SvS]}
  ∪ list_to_set
      (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1) set_solver.
      }
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
{[svar_fresh_s SvS]}
∪ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ list_to_set (svar_fresh_seq SvS (S n1 `max` S n2))
      {
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
{[svar_fresh_s SvS]}
∪ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ list_to_set (svar_fresh_seq SvS (S n1 `max` S n2))
        simpl.
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
{[svar_fresh_s SvS]}
∪ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ {[svar_fresh_s SvS]}
  ∪ list_to_set
      (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS)
         (n1 `max` n2))
        cut (@list_to_set svar SVarSet _ _ _ (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
        ⊆ list_to_set (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) (n1 `max` n2))).
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
list_to_set
  (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS)
       (n1 `max` n2))
→ {[svar_fresh_s SvS]}
  ∪ list_to_set
      (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
  ⊆ {[svar_fresh_s SvS]}
    ∪ list_to_set
        (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS)
           (n1 `max` n2))
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
list_to_set
  (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS)
       (n1 `max` n2))
        {
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
list_to_set
  (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS)
       (n1 `max` n2))
→ {[svar_fresh_s SvS]}
  ∪ list_to_set
      (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
  ⊆ {[svar_fresh_s SvS]}
    ∪ list_to_set
        (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS)
           (n1 `max` n2))
          set_solver.
        }
Σ: Signature
n1: nat
IHn1: ∀ (SvS : SVarSet) (n2 : nat),
  list_to_set (svar_fresh_seq SvS n1)
  ⊆ list_to_set
      (svar_fresh_seq SvS (n1 `max` n2))
SvS: SVarSet
n2: nat
list_to_set
  (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS)
       (n1 `max` n2))
        specialize (IHn1 ({[svar_fresh_s SvS]} ∪ SvS) n2).
Σ: Signature
n1: nat
SvS: SVarSet
n2: nat
IHn1: list_to_set
  (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS)
     n1)
⊆ list_to_set
    (svar_fresh_seq
       ({[svar_fresh_s SvS]} ∪ SvS)
       (n1 `max` n2))
list_to_set
  (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS) n1)
⊆ list_to_set
    (svar_fresh_seq ({[svar_fresh_s SvS]} ∪ SvS)
       (n1 `max` n2))
        apply IHn1.
      }
    }
  Qed.

  Lemma mmdoeip_svar_open depth E n X ψ:
    maximal_mu_depth_of_evar_in_pattern' depth E (ψ^{svar: n ↦ X})
    = maximal_mu_depth_of_evar_in_pattern' depth E ψ.
Σ: Signature
depth: nat
E: evar
n: db_index
X: svar
ψ: Pattern
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^{svar:n↦X} =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
  Proof.
Σ: Signature
depth: nat
E: evar
n: db_index
X: svar
ψ: Pattern
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^{svar:n↦X} =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
    unfold svar_open.
Σ: Signature
depth: nat
E: evar
n: db_index
X: svar
ψ: Pattern
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[svar:n↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
    move: depth n.
Σ: Signature
E: evar
X: svar
ψ: Pattern
∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
    induction ψ; intros depth n'; simpl; try reflexivity.
Σ: Signature
E: evar
X: svar
n: db_index
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = 0
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' 
  (S depth) E ψ^[svar:S n'↦
  patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
    {
Σ: Signature
E: evar
X: svar
n: db_index
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = 0
      case_match; simpl; try reflexivity.
    }
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' 
  (S depth) E ψ^[svar:S n'↦
  patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
    {
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
      rewrite IHψ1.
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 rewrite IHψ2.
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 reflexivity.
    }
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' 
  (S depth) E ψ^[svar:S n'↦
  patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
    {
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[svar:n'↦patt_free_svar X]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
      rewrite IHψ1.
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 rewrite IHψ2.
Σ: Signature
E: evar
X: svar
ψ1, ψ2: Pattern
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 reflexivity.
    }
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[svar:n↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[svar:n↦patt_free_svar X] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' 
  (S depth) E ψ^[svar:S n'↦
  patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
    {
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[svar:n↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[svar:n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
      rewrite IHψ.
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[svar:n↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E ψ =
maximal_mu_depth_of_evar_in_pattern' depth E ψ reflexivity.
    }
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[svar:n↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' (S depth) E
  ψ^[svar:S n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
    {
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[svar:n↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' (S depth) E
  ψ^[svar:S n'↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
      rewrite IHψ.
Σ: Signature
E: evar
X: svar
ψ: Pattern
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[svar:n↦patt_free_svar X] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ reflexivity.
    }
  Qed.

  Lemma mmdoeip_evar_open depth E n x ψ:
    E <> x ->
    maximal_mu_depth_of_evar_in_pattern' depth E (ψ^{evar: n ↦ x})
    = maximal_mu_depth_of_evar_in_pattern' depth E ψ.
Σ: Signature
depth: nat
E: evar
n: db_index
x: evar
ψ: Pattern
E ≠ x
→ maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^{evar:n↦x} =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
  Proof.
Σ: Signature
depth: nat
E: evar
n: db_index
x: evar
ψ: Pattern
E ≠ x
→ maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^{evar:n↦x} =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
    intros Hne.
Σ: Signature
depth: nat
E: evar
n: db_index
x: evar
ψ: Pattern
Hne: E ≠ x
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^{evar:n↦x} =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
    unfold evar_open.
Σ: Signature
depth: nat
E: evar
n: db_index
x: evar
ψ: Pattern
Hne: E ≠ x
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[evar:n↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
    move: depth n.
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
    induction ψ; intros depth n'; simpl; try reflexivity.
Σ: Signature
E, x: evar
n: db_index
Hne: E ≠ x
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = 0
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
Hne: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
Hne: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[evar:S n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' 
  (S depth) E ψ^[evar:n'↦
  patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
    {
Σ: Signature
E, x: evar
n: db_index
Hne: E ≠ x
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = 0
      case_match; simpl; try reflexivity.
Σ: Signature
E, x: evar
n: db_index
Hne: E ≠ x
depth: nat
n': db_index
e: n = n'
H: compare_nat n n' = Nat_equal n n' e
(if decide (x = E) then depth else 0) = 0
      case_match; simpl; try reflexivity.
Σ: Signature
E, x: evar
n: db_index
Hne: E ≠ x
depth: nat
n': db_index
e: n = n'
H: compare_nat n n' = Nat_equal n n' e
e0: x = E
H0: decide (x = E) = left e0
depth = 0
      subst.
Σ: Signature
E: evar
Hne: E ≠ E
depth: nat
n': db_index
H: compare_nat n' n' = Nat_equal n' n' erefl
H0: decide (E = E) = left erefl
depth = 0 contradiction.
    }
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
Hne: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
Hne: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[evar:S n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' 
  (S depth) E ψ^[evar:n'↦
  patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
    {
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
Hne: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
      rewrite IHψ1.
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
Hne: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 rewrite IHψ2.
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
Hne: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 reflexivity.
    }
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
Hne: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[evar:S n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' 
  (S depth) E ψ^[evar:n'↦
  patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
    {
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
Hne: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ1^[evar:n'↦patt_free_evar x]
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2
      rewrite IHψ1.
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
Hne: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E
        ψ2^[evar:n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 rewrite IHψ2.
Σ: Signature
E, x: evar
ψ1, ψ2: Pattern
Hne: E ≠ x
IHψ1: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ1
IHψ2: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ2
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 reflexivity.
    }
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[evar:n↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[evar:S n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
  maximal_mu_depth_of_evar_in_pattern' depth E
    ψ^[evar:n↦patt_free_evar x] =
  maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' 
  (S depth) E ψ^[evar:n'↦
  patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
    {
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[evar:n↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E
  ψ^[evar:S n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
      rewrite IHψ.
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[evar:n↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' depth E ψ =
maximal_mu_depth_of_evar_in_pattern' depth E ψ reflexivity.
    }
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[evar:n↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' (S depth) E
  ψ^[evar:n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
    {
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[evar:n↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' (S depth) E
  ψ^[evar:n'↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
      rewrite IHψ.
Σ: Signature
E, x: evar
ψ: Pattern
Hne: E ≠ x
IHψ: ∀ (depth : nat) (n : db_index),
maximal_mu_depth_of_evar_in_pattern' depth E ψ^[evar:n↦patt_free_evar x] =
maximal_mu_depth_of_evar_in_pattern' depth E ψ
depth: nat
n': db_index
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ reflexivity.
    }
  Qed.


  Lemma mmdoeip_notin E ψ depth:
    E ∉ free_evars ψ ->
    maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0.
Σ: Signature
E: evar
ψ: Pattern
depth: nat
E ∉ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
  Proof.
Σ: Signature
E: evar
ψ: Pattern
depth: nat
E ∉ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
    intros Hnotin.
Σ: Signature
E: evar
ψ: Pattern
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
    move: E depth Hnotin.
Σ: Signature
ψ: Pattern
∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
    induction ψ; intros E depth Hnotin; simpl in *; try reflexivity.
Σ: Signature
x, E: evar
depth: nat
Hnotin: E ∉ {[x]}
(if decide (x = E) then depth else 0) = 0
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
0
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern' depth E
      ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern' depth E
      ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ = 0
    {
Σ: Signature
x, E: evar
depth: nat
Hnotin: E ∉ {[x]}
(if decide (x = E) then depth else 0) = 0 case_match.
Σ: Signature
x, E: evar
depth: nat
Hnotin: E ∉ {[x]}
e: x = E
H: decide (x = E) = left e
depth = 0
Σ: Signature
x, E: evar
depth: nat
Hnotin: E ∉ {[x]}
n: x ≠ E
H: decide (x = E) = right n
0 = 0 set_solver.
Σ: Signature
x, E: evar
depth: nat
Hnotin: E ∉ {[x]}
n: x ≠ E
H: decide (x = E) = right n
0 = 0 reflexivity. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
0
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern' depth E
      ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern' depth E
      ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ = 0
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
0 rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
0 set_solver.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
0 rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0 `max` 0 = 0 set_solver.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0 `max` 0 = 0 reflexivity. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern' depth E
      ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern' depth E
      ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ = 0
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' depth E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
0 rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
0 set_solver.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0
`max` maximal_mu_depth_of_evar_in_pattern' depth E ψ2 =
0 rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0 `max` 0 = 0 set_solver.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ1 = 0
IHψ2: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern' depth
      E ψ2 = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ1 ∪ free_evars ψ2
0 `max` 0 = 0 reflexivity. }
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern' depth E
      ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ = 0
    {
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0 rewrite IHψ.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
E ∉ free_evars ψ
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern' depth E
      ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
0 = 0 exact Hnotin.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
0 = 0 reflexivity. }
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ = 0
    {
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ = 0 rewrite IHψ.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
E ∉ free_evars ψ
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∉ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern' depth E
      ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
0 = 0 exact Hnotin.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∉ free_evars ψ → maximal_mu_depth_of_evar_in_pattern' depth E ψ = 0
E: evar
depth: nat
Hnotin: E ∉ free_evars ψ
0 = 0 reflexivity. }
  Qed.

  Lemma mmdoeip_S_in E ψ depth:
    E ∈ free_evars ψ ->
    maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ
    = S (maximal_mu_depth_of_evar_in_pattern' depth E ψ).
Σ: Signature
E: evar
ψ: Pattern
depth: nat
E ∈ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
  S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
  Proof.
Σ: Signature
E: evar
ψ: Pattern
depth: nat
E ∈ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
  S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
    intros Hin.
Σ: Signature
E: evar
ψ: Pattern
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
    move: E depth Hin.
Σ: Signature
ψ: Pattern
∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
    S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
    induction ψ; intros E depth Hin; simpl in *.
Σ: Signature
x, E: evar
depth: nat
Hin: E ∈ {[x]}
(if decide (x = E) then S depth else 0) =
S (if decide (x = E) then depth else 0)
Σ: Signature
x: svar
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S (S depth)) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ)
    {
Σ: Signature
x, E: evar
depth: nat
Hin: E ∈ {[x]}
(if decide (x = E) then S depth else 0) =
S (if decide (x = E) then depth else 0) case_match.
Σ: Signature
x, E: evar
depth: nat
Hin: E ∈ {[x]}
e: x = E
H: decide (x = E) = left e
S depth = S depth
Σ: Signature
x, E: evar
depth: nat
Hin: E ∈ {[x]}
n: x ≠ E
H: decide (x = E) = right n
0 = 1 reflexivity.
Σ: Signature
x, E: evar
depth: nat
Hin: E ∈ {[x]}
n: x ≠ E
H: decide (x = E) = right n
0 = 1 set_solver. }
Σ: Signature
x: svar
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S (S depth)) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ)
    {
Σ: Signature
x: svar
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1 set_solver. }
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S (S depth)) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ)
    {
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1 set_solver. }
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S (S depth)) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ)
    {
Σ: Signature
n: db_index
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1 set_solver. }
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S (S depth)) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ)
    {
Σ: Signature
sigma: symbols
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1 set_solver. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S (S depth)) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ)
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      destruct (decide (E ∈ free_evars ψ1)),(decide (E ∈ free_evars ψ2)).
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
        rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
E ∈ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
E ∈ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) reflexivity.
      }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∈ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      rewrite [maximal_mu_depth_of_evar_in_pattern' _ _ ψ2]mmdoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      rewrite [maximal_mu_depth_of_evar_in_pattern' _ _ ψ2]mmdoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
      simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
      rewrite Nat.max_comm.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1) =
S
  (0
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ1)
      simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1) =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
      reflexivity.
    }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∈ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      rewrite [maximal_mu_depth_of_evar_in_pattern' _ _ ψ1]mmdoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      rewrite [maximal_mu_depth_of_evar_in_pattern' _ _ ψ1]mmdoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (0
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (0
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ2) =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ2)
      reflexivity.
    }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      exfalso.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
False set_solver.
    }
  }
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S (S depth)) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ)
  {
Σ: Signature
E: evar
depth: nat
Hin: E ∈ ∅
0 = 1 set_solver. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S (S depth)) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ)
  {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    destruct (decide (E ∈ free_evars ψ1)),(decide (E ∈ free_evars ψ2)).
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
      rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
E ∈ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
E ∈ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
e0: E ∈ free_evars ψ2
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) reflexivity.
    }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    rewrite IHψ1.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∈ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    rewrite [maximal_mu_depth_of_evar_in_pattern' _ _ ψ2]mmdoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    rewrite [maximal_mu_depth_of_evar_in_pattern' _ _ ψ2]mmdoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
E ∉ free_evars ψ2 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
`max` 0 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
    simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` 0)
    rewrite Nat.max_comm.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1) =
S
  (0
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ1)
    simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
e: E ∈ free_evars ψ1
n: E ∉ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1) =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ1)
    reflexivity.
  }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' 
        (S depth) E ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
  {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    rewrite IHψ2.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∈ free_evars ψ2
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) assumption.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    rewrite [maximal_mu_depth_of_evar_in_pattern' _ _ ψ1]mmdoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    rewrite [maximal_mu_depth_of_evar_in_pattern' _ _ ψ1]mmdoeip_notin.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (0
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
E ∉ free_evars ψ1 assumption. }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
0
`max` S
        (maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2) =
S
  (0
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    simpl.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
e: E ∈ free_evars ψ2
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ2) =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ2)
    reflexivity.
  }
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
  {
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ1
`max` maximal_mu_depth_of_evar_in_pattern' (S depth) E
        ψ2 =
S
  (maximal_mu_depth_of_evar_in_pattern' depth E ψ1
   `max` maximal_mu_depth_of_evar_in_pattern' depth E
           ψ2)
    exfalso.
Σ: Signature
ψ1, ψ2: Pattern
IHψ1: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ1
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ1 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ1)
IHψ2: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ2
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ2 =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ2)
E: evar
depth: nat
Hin: E ∈ free_evars ψ1 ∪ free_evars ψ2
n: E ∉ free_evars ψ1
n0: E ∉ free_evars ψ2
False set_solver.
  }
}
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S (S depth)) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ)
{
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
  rewrite IHψ.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
E ∈ free_evars ψ
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ) =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ) assumption.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ) =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ) reflexivity.
}
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S (S depth)) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ)
{
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
maximal_mu_depth_of_evar_in_pattern' (S (S depth)) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ)
  rewrite IHψ.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
E ∈ free_evars ψ
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
  E ∈ free_evars ψ
  → maximal_mu_depth_of_evar_in_pattern'
      (S depth) E ψ =
    S
      (maximal_mu_depth_of_evar_in_pattern'
         depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ) =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ) assumption.
Σ: Signature
ψ: Pattern
IHψ: ∀ (E : evar) (depth : nat),
E ∈ free_evars ψ
→ maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ =
S (maximal_mu_depth_of_evar_in_pattern' depth E ψ)
E: evar
depth: nat
Hin: E ∈ free_evars ψ
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ) =
S (maximal_mu_depth_of_evar_in_pattern' (S depth) E ψ) reflexivity.
}
Qed.


  Lemma congruence_ex_helper Γ E ψ x p q gpi exdepth mudepth EvS SvS
    (HxneqE : x ≠ E)
    (wfψ : well_formed (ex , ψ))
    (wfp : well_formed p)
    (wfq : well_formed q)
    (Heqx : x = evar_fresh (elements EvS))
    (HEinψ: E ∈ free_evars ψ)
    (i': ProofInfo)
    (p_sub_EvS: free_evars p ⊆ EvS)
    (q_sub_EvS: free_evars q ⊆ EvS)
    (ψ_sub_EvS : free_evars ψ ⊆ EvS)
    (Heqi': i' =
            (ExGen := list_to_set
                        (evar_fresh_seq EvS
                           (maximal_exists_depth_of_evar_in_pattern' exdepth E
                              (ex , ψ))),
             SVSubst := list_to_set
                          (svar_fresh_seq SvS
                             (maximal_mu_depth_of_evar_in_pattern' mudepth E
                                (ex , ψ))),
             KT := (if
                     decide
                       (0 =
                        maximal_mu_depth_of_evar_in_pattern' mudepth E (ex , ψ))
                    is left _ then false
                    else true),
             FP := ∅
            ))
    (pile: ProofInfoLe i' ( gpi))
    (IH: Γ ⊢i ψ^{evar: 0 ↦ x}^[[evar: E ↦ p]] <---> ψ^{evar: 0 ↦ x}^[[evar: E ↦ q]]
       using  gpi) :
    (Γ ⊢i (ex , ψ^[[evar: E ↦ p]]) <---> (ex , ψ^[[evar: E ↦ q]]) using  gpi).
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
  Proof.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
    apply pf_evar_open_free_evar_subst_equiv_sides in IH.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
x ≠ E
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
well_formed p = true
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
well_formed q = true
    2: {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
x ≠ E exact HxneqE. }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
well_formed p = true
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
well_formed q = true
    2: {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
well_formed p = true exact wfp. }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
well_formed q = true
    2: {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
well_formed q = true exact wfq. }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
    unshelve (epose proof (IH1 := pf_iff_proj1 Γ _ _ _ _ _ IH)).
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦p]]^{evar:0↦x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦q]]^{evar:0↦x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦p]]^{evar:0↦x} abstract (wf_auto2). }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦q]]^{evar:0↦x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦q]]^{evar:0↦x} abstract (wf_auto2). }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
    unshelve (epose proof (IH2 := pf_iff_proj2 Γ _ _ _ _ _ IH)).
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦p]]^{evar:0↦x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦q]]^{evar:0↦x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦p]]^{evar:0↦x} abstract (wf_auto2). }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦q]]^{evar:0↦x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦q]]^{evar:0↦x} abstract (wf_auto2). }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi

    (* TODO: remove the well-formedness constraints on this lemma*)
    apply pf_iff_split.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]])
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]])
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) ---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦q]]) ---> (ex , ψ^[[evar:E↦p]])
using gpi
    {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]]) abstract (wf_auto2). }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]])
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) ---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦q]]) ---> (ex , ψ^[[evar:E↦p]])
using gpi
    {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]]) abstract (wf_auto2). }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) ---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦q]]) ---> (ex , ψ^[[evar:E↦p]])
using gpi
    {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) ---> (ex , ψ^[[evar:E↦q]])
using gpi
      eapply strip_exists_quantify_l.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
?x ∉ free_evars ψ^[[evar:E↦p]]
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed_closed_ex_aux ψ^[[evar:E↦p]] 1
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i exists_quantify ?x ψ^[[evar:E↦p]]^{evar:0↦?x} --->
   (ex , ψ^[[evar:E↦q]]) using gpi
      3: {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i exists_quantify ?x ψ^[[evar:E↦p]]^{evar:0↦?x} --->
   (ex , ψ^[[evar:E↦q]]) using gpi
        apply Ex_gen.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
ProofInfoLe
  (ExGen := {[?x]}, SVSubst := ∅, KT := false, FP := ∅)
  gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
?x ∉ free_evars (ex , ψ^[[evar:E↦q]])
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦
   ?x} ---> (ex , ψ^[[evar:E↦q]]) using gpi
        3: {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦?x} --->
   (ex , ψ^[[evar:E↦q]]) using gpi
          eapply syllogism_meta.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦p]]^{evar:0↦?x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed ?B
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]])
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i ψ^[[evar:E↦p]]^{evar:0↦?x} ---> ?B using gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i ?B ---> (ex , ψ^[[evar:E↦q]]) using gpi
          5: {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i ?B ---> (ex , ψ^[[evar:E↦q]]) using gpi
            useBasicReasoning.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i ?B ---> (ex , ψ^[[evar:E↦q]])
using BasicReasoning
            apply Ex_quan.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]])
            abstract (wf_auto2).
          }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦p]]^{evar:0↦?x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed
  (ex , ψ^[[evar:E↦q]]) ^ [patt_free_evar ?y]
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]])
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦
   ?x} --->
   (ex , ψ^[[evar:E↦q]]) ^ [patt_free_evar ?y]
using gpi
          4: {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦?x} --->
   (ex , ψ^[[evar:E↦q]]) ^ [patt_free_evar ?y]
using gpi
              apply IH1.
            }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦p]]^{evar:0↦x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]]) ^ [patt_free_evar x]
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]])
          {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦p]]^{evar:0↦x} abstract (wf_auto2). }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]]) ^ [patt_free_evar x]
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]])
          {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]]) ^ [patt_free_evar x] abstract (simpl; wf_auto2; apply wfc_ex_aux_bevar_subst; wf_auto2). }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]])
          {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦q]]) abstract (wf_auto2). }
        }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
ProofInfoLe
  (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := ∅)
  gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
x ∉ free_evars (ex , ψ^[[evar:E↦q]])
        {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
ProofInfoLe
  (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := ∅)
  gpi
          abstract (
            subst i';
            eapply pile_trans;
            [|apply pile];
            apply pile_evs_svs_kt;
            [(
            simpl;
            rewrite medoeip_S_in;
            [assumption|];
            simpl;
            unfold evar_fresh_s;
            rewrite -Heqx;
            clear;
            set_solver
            )|(clear; set_solver)
            |reflexivity|(clear; set_solver)]
          ).
        }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
x ∉ free_evars (ex , ψ^[[evar:E↦q]])
        {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
x ∉ free_evars (ex , ψ^[[evar:E↦q]])
          abstract (
            simpl;
            pose proof (Htmp1 := set_evar_fresh_is_fresh' EvS);
            pose proof (Htmp2 := free_evars_free_evar_subst ψ q E);
            set_solver
          ).
        }
      }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
x ∉ free_evars ψ^[[evar:E↦p]]
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed_closed_ex_aux ψ^[[evar:E↦p]] 1
      {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
x ∉ free_evars ψ^[[evar:E↦p]]
        abstract (
          simpl;
          pose proof (Htmp1 := @set_evar_fresh_is_fresh' _ EvS);
          pose proof (Htmp := @free_evars_free_evar_subst Σ ψ p E);
          set_solver
        ).
      }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed_closed_ex_aux ψ^[[evar:E↦p]] 1
      {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed_closed_ex_aux ψ^[[evar:E↦p]] 1
        abstract (wf_auto2).
      }
    }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦q]]) ---> (ex , ψ^[[evar:E↦p]])
using gpi
    (* this block is a symmetric version of the previous block*)
    {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i (ex , ψ^[[evar:E↦q]]) ---> (ex , ψ^[[evar:E↦p]])
using gpi
      eapply strip_exists_quantify_l.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
?x ∉ free_evars ψ^[[evar:E↦q]]
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed_closed_ex_aux ψ^[[evar:E↦q]] 1
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i exists_quantify ?x ψ^[[evar:E↦q]]^{evar:0↦?x} --->
   (ex , ψ^[[evar:E↦p]]) using gpi
      3: {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i exists_quantify ?x ψ^[[evar:E↦q]]^{evar:0↦?x} --->
   (ex , ψ^[[evar:E↦p]]) using gpi
        apply Ex_gen.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
ProofInfoLe
  (ExGen := {[?x]}, SVSubst := ∅, KT := false, FP := ∅)
  gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
?x ∉ free_evars (ex , ψ^[[evar:E↦p]])
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦
   ?x} ---> (ex , ψ^[[evar:E↦p]]) using gpi
        3: {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦?x} --->
   (ex , ψ^[[evar:E↦p]]) using gpi
          eapply syllogism_meta.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦q]]^{evar:0↦?x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed ?B
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]])
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i ψ^[[evar:E↦q]]^{evar:0↦?x} ---> ?B using gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i ?B ---> (ex , ψ^[[evar:E↦p]]) using gpi
          5: {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i ?B ---> (ex , ψ^[[evar:E↦p]]) using gpi
            useBasicReasoning.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ ⊢i ?B ---> (ex , ψ^[[evar:E↦p]])
using BasicReasoning
            apply Ex_quan.
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]])
            abstract (wf_auto2).
          }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦q]]^{evar:0↦?x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed
  (ex , ψ^[[evar:E↦p]]) ^ [patt_free_evar ?y]
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]])
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦
   ?x} --->
   (ex , ψ^[[evar:E↦p]]) ^ [patt_free_evar ?y]
using gpi
          4: {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦?x} --->
   (ex , ψ^[[evar:E↦p]]) ^ [patt_free_evar ?y]
using gpi
              apply IH2.
            }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦q]]^{evar:0↦x}
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]]) ^ [patt_free_evar x]
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]])
          {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed ψ^[[evar:E↦q]]^{evar:0↦x} abstract (wf_auto2). }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]]) ^ [patt_free_evar x]
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]])
          {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]]) ^ [patt_free_evar x] abstract (simpl; wf_auto2; apply wfc_ex_aux_bevar_subst; wf_auto2). }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]])
          {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed (ex , ψ^[[evar:E↦p]]) abstract (wf_auto2). }
        }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
ProofInfoLe
  (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := ∅)
  gpi
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
x ∉ free_evars (ex , ψ^[[evar:E↦p]])
        {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
ProofInfoLe
  (ExGen := {[x]}, SVSubst := ∅, KT := false, FP := ∅)
  gpi
          abstract (
            subst i';
            eapply pile_trans;
            [|apply pile];
            apply pile_evs_svs_kt;
            [(
              simpl;
              rewrite medoeip_S_in;
              [assumption|];
              simpl;
              unfold evar_fresh_s;
              rewrite -Heqx;
              clear;
              set_solver
            )
           |(clear; set_solver)
           |(reflexivity)|(clear; set_solver)
            ]).
        }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
x ∉ free_evars (ex , ψ^[[evar:E↦p]])
        {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
x ∉ free_evars (ex , ψ^[[evar:E↦p]])
          abstract (
            simpl;
            pose proof (Htmp1 := set_evar_fresh_is_fresh' EvS);
            pose proof (Htmp := free_evars_free_evar_subst ψ p E);
            set_solver
          ).
        }
      }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
x ∉ free_evars ψ^[[evar:E↦q]]
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed_closed_ex_aux ψ^[[evar:E↦q]] 1
      {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
x ∉ free_evars ψ^[[evar:E↦q]]
        abstract (
          simpl;
          pose proof (Htmp1 := set_evar_fresh_is_fresh' EvS);
          pose proof (Htmp := free_evars_free_evar_subst ψ q E);
          set_solver
        ).
      }
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed_closed_ex_aux ψ^[[evar:E↦q]] 1
      {
Σ: Signature
Γ: Theory
E: evar
ψ: Pattern
x: evar
p, q: Pattern
gpi: ProofInfo
exdepth, mudepth: nat
EvS: EVarSet
SvS: SVarSet
HxneqE: x ≠ E
wfψ: well_formed (ex , ψ)
wfp: well_formed p
wfq: well_formed q
Heqx: x = evar_fresh (elements EvS)
HEinψ: E ∈ free_evars ψ
i': ProofInfo
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
Heqi': i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
pile: ProofInfoLe i' gpi
IH: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} <--->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^{evar:0↦x} --->
   ψ^[[evar:E↦q]]^{evar:0↦x} using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^{evar:0↦x} --->
   ψ^[[evar:E↦p]]^{evar:0↦x} using gpi
well_formed_closed_ex_aux ψ^[[evar:E↦q]] 1
        abstract (wf_auto2).
      }
    }
  Defined.

  Equations? frames_on_the_way_to_hole'
  (EvS : EVarSet)
  (SvS : SVarSet)
  (E : evar)
  (ψ p q : Pattern)
  (wfψ : well_formed ψ = true)
  (wfp : well_formed p = true)
  (wfq : well_formed q = true)
  : gset wfPattern
  by wf (size' ψ) lt :=
  @frames_on_the_way_to_hole' EvS SvS E (patt_free_evar _) _ _ _ _ _ := ∅ ;

  @frames_on_the_way_to_hole' EvS SvS E (patt_free_svar _) _ _ _ _ _ := ∅ ;

  @frames_on_the_way_to_hole' EvS SvS E (patt_bound_evar _) _ _ _ _ _ := ∅ ;

  @frames_on_the_way_to_hole' EvS SvS E (patt_bound_svar _) _ _ _ _ _ := ∅ ;

  @frames_on_the_way_to_hole' EvS SvS E (patt_sym _) _ _ _ _ _ := ∅ ;

  @frames_on_the_way_to_hole' EvS SvS E (patt_bott) _ _ _ _ _ := ∅ ;

  @frames_on_the_way_to_hole' EvS SvS E (patt_app ψ1 ψ2) p q _ _ _
    :=
      (@singleton _ _ gset_singleton (exist _ (ψ1^[[evar: E ↦ p]]) _))
      ∪ {[(exist _ (ψ2^[[evar: E ↦ p]]) _)]}
      ∪ {[(exist _ (ψ1^[[evar: E ↦ q]]) _)]}
      ∪ {[(exist _ (ψ2^[[evar: E ↦ q]]) _)]}
      ∪ (@frames_on_the_way_to_hole' EvS SvS E ψ1 p q _ _ _)
      ∪ (@frames_on_the_way_to_hole' EvS SvS E ψ2 p q _ _ _) ;

  @frames_on_the_way_to_hole' EvS SvS E (patt_imp ψ1 ψ2) p q _ _ _
  := @union _ gset_union (@frames_on_the_way_to_hole' EvS SvS E ψ1 p q _ _ _)
     (@frames_on_the_way_to_hole' EvS SvS E ψ2 p q _ _ _) ;

  @frames_on_the_way_to_hole' EvS SvS E (patt_exists ψ') p q _ _ _
   := (@frames_on_the_way_to_hole' ({[(evar_fresh (elements EvS))]} ∪ EvS) SvS E (ψ'^{evar: 0 ↦ evar_fresh (elements EvS)}) p q _ _ _) ;

  @frames_on_the_way_to_hole' EvS SvS E (patt_mu ψ') _ _ _ _ _
   := (@frames_on_the_way_to_hole' EvS ({[(svar_fresh (elements SvS))]} ∪ SvS) E (ψ'^{svar: 0 ↦ svar_fresh (elements SvS)}) p q _ _ _)
  .
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ1^[[evar:E↦p]] = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ2^[[evar:E↦p]] = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ1^[[evar:E↦q]] = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ2^[[evar:E↦q]] = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ1 = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ1 < size' (ψ1 $ ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ2 = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ2 < size' (ψ1 $ ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ1 = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ1 < size' (ψ1 ---> ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ2 = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ2 < size' (ψ1 ---> ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ex , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ'^{evar:0↦evar_fresh (elements EvS)} =
true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ex , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ex , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ex , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{evar:0↦evar_fresh (elements EvS)} <
size' (ex , ψ')
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (mu , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ'^{svar:0↦svar_fresh (elements SvS)} =
true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (mu , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (mu , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (mu , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{svar:0↦svar_fresh (elements SvS)} <
size' (mu , ψ')
  Proof.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ1^[[evar:E↦p]] = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ2^[[evar:E↦p]] = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ1^[[evar:E↦q]] = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ2^[[evar:E↦q]] = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ1 = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ1 < size' (ψ1 $ ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ2 = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ2 < size' (ψ1 $ ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ1 = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ1 < size' (ψ1 ---> ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ2 = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ2 < size' (ψ1 ---> ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ex , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ'^{evar:0↦evar_fresh (elements EvS)} =
true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ex , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ex , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ex , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{evar:0↦evar_fresh (elements EvS)} <
size' (ex , ψ')
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (mu , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed ψ'^{svar:0↦svar_fresh (elements SvS)} =
true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (mu , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed p = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (mu , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
well_formed q = true
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (mu , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{svar:0↦svar_fresh (elements SvS)} <
size' (mu , ψ')
    all: try (abstract (solve [try_wfauto2])).
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ1 < size' (ψ1 $ ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 $ ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ2 < size' (ψ1 $ ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ1 < size' (ψ1 ---> ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ψ1 ---> ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ2 < size' (ψ1 ---> ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (ex , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{evar:0↦evar_fresh (elements EvS)} <
size' (ex , ψ')
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            size'
             (mu , ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{svar:0↦svar_fresh (elements SvS)} <
size' (mu , ψ')
    all: simpl in *.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S
             (size' ψ1 +
             size' ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ1 < S (size' ψ1 + size' ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S
             (size' ψ1 +
             size' ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ2 < S (size' ψ1 + size' ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S
             (size' ψ1 +
             size' ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ1 < S (size' ψ1 + size' ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S
             (size' ψ1 +
             size' ψ2)
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ2 < S (size' ψ1 + size' ψ2)
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S (size' ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{evar:0↦evar_fresh (elements EvS)} <
S (size' ψ')
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S (size' ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{svar:0↦svar_fresh (elements SvS)} <
S (size' ψ')
    all: try abstract(lia).
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S (size' ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{evar:0↦evar_fresh (elements EvS)} <
S (size' ψ')
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S (size' ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{svar:0↦svar_fresh (elements SvS)} <
S (size' ψ')
    {
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S (size' ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{evar:0↦evar_fresh (elements EvS)} <
S (size' ψ') rewrite evar_open_size'.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (ex , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S (size' ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ' < S (size' ψ') abstract(lia). }
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S (size' ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{svar:0↦svar_fresh (elements SvS)} <
S (size' ψ')
    {
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S (size' ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ'^{svar:0↦svar_fresh (elements SvS)} <
S (size' ψ') rewrite svar_open_size'.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ', p, q: Pattern
wfψ: well_formed (mu , ψ') = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole': EVarSet
→ SVarSet
  → evar
    → ∀ x2 x3
         x4 : Pattern,
        well_formed x2 =
        true
        → well_formed x3 =
          true
          → well_formed
             x4 = true
            → 
            size' x2 <
            S (size' ψ')
            → 
            mapset.mapset'
             (gmap
             wfPattern ())
size' ψ' < S (size' ψ') abstract(lia). }
  Defined.

End FOL_helpers.

  Ltac pi_exact H := 
    lazymatch type of H with
    | ?H' =>
      lazymatch goal with
      | [|- ?g] =>
        (cut (H' = g);
        [(let H0 := fresh "H0" in intros H0; rewrite -H0; exact H)|
         (repeat f_equal; try reflexivity; try apply proof_irrel)])
      end
    end.

  Ltac pi_assumption :=
    match goal with
    | [H : _ |- _] => pi_exact H
    end.

  Ltac pi_set_solver := set_solver by (try pi_assumption).

Section FOL_helpers.

  Context {Σ : Signature}.

  Lemma frames_on_the_way_to_hole'_app_1 EvS SvS E ψ1 ψ2 p q wfψ1 wfψ wfp wfq :
    (@frames_on_the_way_to_hole' Σ EvS SvS E ψ1 p q wfψ1 wfp wfq)
    ⊆
    (@frames_on_the_way_to_hole' Σ EvS SvS E (ψ1 $ ψ2) p q wfψ wfp wfq).
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ1: well_formed ψ1 = true
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ1 p q wfψ1 wfp
  wfq
⊆ frames_on_the_way_to_hole' EvS SvS E (ψ1 $ ψ2) p q
    wfψ wfp wfq
  Proof.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ1: well_formed ψ1 = true
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ1 p q wfψ1 wfp
  wfq
⊆ frames_on_the_way_to_hole' EvS SvS E (ψ1 $ ψ2) p q
    wfψ wfp wfq
    simp frames_on_the_way_to_hole'.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ1: well_formed ψ1 = true
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ1 p q wfψ1 wfp
  wfq
⊆ {[ψ1^[[evar:E↦p]]
    ↾ frames_on_the_way_to_hole'_obligation_1 E ψ1 ψ2
        p wfψ wfp;
  ψ2^[[evar:E↦p]]
  ↾ frames_on_the_way_to_hole'_obligation_2 E ψ1 ψ2 p
      wfψ wfp;
  ψ1^[[evar:E↦q]]
  ↾ frames_on_the_way_to_hole'_obligation_3 E ψ1 ψ2 q
      wfψ wfq;
  ψ2^[[evar:E↦q]]
  ↾ frames_on_the_way_to_hole'_obligation_4 E ψ1 ψ2 q
      wfψ wfq]}
  ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
      (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_6 p wfp)
      (frames_on_the_way_to_hole'_obligation_7 q wfq)
  ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
    pi_set_solver.
  Qed.

  Lemma frames_on_the_way_to_hole'_app_2 EvS SvS E ψ1 ψ2 p q wfψ2 wfψ wfp wfq:
    (@frames_on_the_way_to_hole' Σ EvS SvS E ψ2 p q wfψ2 wfp wfq)
    ⊆
    (@frames_on_the_way_to_hole' Σ EvS SvS E (ψ1 $ ψ2) p q wfψ wfp wfq).
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ2: well_formed ψ2 = true
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ2 p q wfψ2 wfp
  wfq
⊆ frames_on_the_way_to_hole' EvS SvS E (ψ1 $ ψ2) p q
    wfψ wfp wfq
  Proof.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ2: well_formed ψ2 = true
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ2 p q wfψ2 wfp
  wfq
⊆ frames_on_the_way_to_hole' EvS SvS E (ψ1 $ ψ2) p q
    wfψ wfp wfq
    simp frames_on_the_way_to_hole'.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ2: well_formed ψ2 = true
wfψ: well_formed (ψ1 $ ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ2 p q wfψ2 wfp
  wfq
⊆ {[ψ1^[[evar:E↦p]]
    ↾ frames_on_the_way_to_hole'_obligation_1 E ψ1 ψ2
        p wfψ wfp;
  ψ2^[[evar:E↦p]]
  ↾ frames_on_the_way_to_hole'_obligation_2 E ψ1 ψ2 p
      wfψ wfp;
  ψ1^[[evar:E↦q]]
  ↾ frames_on_the_way_to_hole'_obligation_3 E ψ1 ψ2 q
      wfψ wfq;
  ψ2^[[evar:E↦q]]
  ↾ frames_on_the_way_to_hole'_obligation_4 E ψ1 ψ2 q
      wfψ wfq]}
  ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
      (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_6 p wfp)
      (frames_on_the_way_to_hole'_obligation_7 q wfq)
  ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
    pi_set_solver.
  Qed.

  Lemma frames_on_the_way_to_hole'_imp_1 EvS SvS E ψ1 ψ2 p q wfψ1 wfψ wfp wfq :
    (@frames_on_the_way_to_hole' Σ EvS SvS E ψ1 p q wfψ1 wfp wfq)
    ⊆
    (@frames_on_the_way_to_hole' Σ EvS SvS E (ψ1 ---> ψ2) p q wfψ wfp wfq).
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ1: well_formed ψ1 = true
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ1 p q wfψ1 wfp
  wfq
⊆ frames_on_the_way_to_hole' EvS SvS E (ψ1 ---> ψ2) p
    q wfψ wfp wfq
  Proof.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ1: well_formed ψ1 = true
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ1 p q wfψ1 wfp
  wfq
⊆ frames_on_the_way_to_hole' EvS SvS E (ψ1 ---> ψ2) p
    q wfψ wfp wfq
    simp frames_on_the_way_to_hole'.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ1: well_formed ψ1 = true
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ1 p q wfψ1 wfp
  wfq
⊆ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
    (frames_on_the_way_to_hole'_obligation_13 ψ1 ψ2
       wfψ)
    (frames_on_the_way_to_hole'_obligation_14 p wfp)
    (frames_on_the_way_to_hole'_obligation_15 q wfq)
  ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_17 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_18 p wfp)
      (frames_on_the_way_to_hole'_obligation_19 q wfq)
    (*unfold frames_on_the_way_to_hole'_unfold_clause_7.*)
    pi_set_solver.
  Qed.

  Lemma frames_on_the_way_to_hole'_imp_2 EvS SvS E ψ1 ψ2 p q wfψ2 wfψ wfp wfq:
    (@frames_on_the_way_to_hole' Σ EvS SvS E ψ2 p q wfψ2 wfp wfq)
    ⊆
    (@frames_on_the_way_to_hole' Σ EvS SvS E (ψ1 ---> ψ2) p q wfψ wfp wfq).
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ2: well_formed ψ2 = true
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ2 p q wfψ2 wfp
  wfq
⊆ frames_on_the_way_to_hole' EvS SvS E (ψ1 ---> ψ2) p
    q wfψ wfp wfq
  Proof.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ2: well_formed ψ2 = true
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ2 p q wfψ2 wfp
  wfq
⊆ frames_on_the_way_to_hole' EvS SvS E (ψ1 ---> ψ2) p
    q wfψ wfp wfq
    simp frames_on_the_way_to_hole'.
Σ: Signature
EvS: EVarSet
SvS: SVarSet
E: evar
ψ1, ψ2, p, q: Pattern
wfψ2: well_formed ψ2 = true
wfψ: well_formed (ψ1 ---> ψ2) = true
wfp: well_formed p = true
wfq: well_formed q = true
frames_on_the_way_to_hole' EvS SvS E ψ2 p q wfψ2 wfp
  wfq
⊆ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
    (frames_on_the_way_to_hole'_obligation_13 ψ1 ψ2
       wfψ)
    (frames_on_the_way_to_hole'_obligation_14 p wfp)
    (frames_on_the_way_to_hole'_obligation_15 q wfq)
  ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_17 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_18 p wfp)
      (frames_on_the_way_to_hole'_obligation_19 q wfq)
    pi_set_solver.
  Qed.

  Lemma helper_app_lemma Γ ψ1 ψ2 p q E i
    (wfψ1: well_formed ψ1)
    (wfψ2: well_formed ψ2)
    (wfp: well_formed p)
    (wfq: well_formed q)
    (pile: ProofInfoLe ( (
      ExGen := ∅, 
      SVSubst := ∅, 
      KT := false, 
      FP := {[(exist _ (ψ1^[[evar: E ↦ p]]) (well_formed_free_evar_subst_0 E _ _ wfp wfψ1))
            ;(exist _ (ψ1^[[evar: E ↦ q]]) (well_formed_free_evar_subst_0 E _ _ wfq wfψ1))
            ;(exist _ (ψ2^[[evar: E ↦ p]]) (well_formed_free_evar_subst_0 E _ _ wfp wfψ2))
            ;(exist _ (ψ2^[[evar: E ↦ q]]) (well_formed_free_evar_subst_0 E _ _ wfq wfψ2))
            ]} )) i )
    (pf₁: Γ ⊢i ψ1^[[evar: E ↦ p]] <---> ψ1^[[evar: E ↦ q]] using i)
    (pf₂: Γ ⊢i ψ2^[[evar: E ↦ p]] <---> ψ2^[[evar: E ↦ q]] using i)
    :
    (Γ ⊢i (ψ1^[[evar: E ↦ p]]) $ (ψ2^[[evar: E ↦ p]]) <---> (ψ1^[[evar: E ↦ q]]) $ (ψ2^[[evar: E ↦ q]]) using i).
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]]
           ↾ well_formed_free_evar_subst_0 E ψ1
               p wfp wfψ1;
         ψ1^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ1 q
             wfq wfψ1;
         ψ2^[[evar:E↦p]]
         ↾ well_formed_free_evar_subst_0 E ψ2 p
             wfp wfψ2;
         ψ2^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ2 q
             wfq wfψ2]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using i
  Proof.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]]
           ↾ well_formed_free_evar_subst_0 E ψ1
               p wfp wfψ1;
         ψ1^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ1 q
             wfq wfψ1;
         ψ2^[[evar:E↦p]]
         ↾ well_formed_free_evar_subst_0 E ψ2 p
             wfp wfψ2;
         ψ2^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ2 q
             wfq wfψ2]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using i
    remember (well_formed_free_evar_subst_0 E _ _ wfp wfψ1) as Hwf1.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ1 q
             wfq wfψ1;
         ψ2^[[evar:E↦p]]
         ↾ well_formed_free_evar_subst_0 E ψ2 p
             wfp wfψ2;
         ψ2^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ2 q
             wfq wfψ2]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using i
    remember (well_formed_free_evar_subst_0 E _ _ wfq wfψ1) as Hwf2.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]]
         ↾ well_formed_free_evar_subst_0 E ψ2 p
             wfp wfψ2;
         ψ2^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ2 q
             wfq wfψ2]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using i
    remember (well_formed_free_evar_subst_0 E _ _ wfp wfψ2) as Hwf3.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ2 q
             wfq wfψ2]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using i
    remember (well_formed_free_evar_subst_0 E _ _ wfq wfψ2) as Hwf4.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using i

    eapply pf_iff_equiv_trans.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed ?B
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <---> ?B
using i
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ?B <---> ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]]
using i
    5: {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ?B <---> ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]]
using i 
      apply conj_intro_meta.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (?B ---> ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] ---> ?B)
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ?B ---> ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using i
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] ---> ?B using i
      4: {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] ---> ?B using i
        eapply Framing_right with (ψ := ψ1^[[evar: E ↦ q]]) (wfψ := Hwf2).
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦q]] ↾ Hwf2]}) i
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ2^[[evar:E↦q]] ---> ?ϕ₂ using i
        1: {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦q]] ↾ Hwf2]}) i eapply pile_trans.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦q]] ↾ Hwf2]}) ?i₂
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe ?i₂ i 2: apply pile.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦q]] ↾ Hwf2]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) try_solve_pile. }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ2^[[evar:E↦q]] ---> ?ϕ₂ using i
        {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ2^[[evar:E↦q]] ---> ?ϕ₂ using i
          eapply pf_conj_elim_r_meta in pf₂.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦q]] ---> ψ2^[[evar:E↦p]]
using i
Γ ⊢i ψ2^[[evar:E↦q]] ---> ?ϕ₂ using i
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦p]] ---> ψ2^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦q]] ---> ψ2^[[evar:E↦p]])
          apply pf₂.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦p]] ---> ψ2^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦q]] ---> ψ2^[[evar:E↦p]])
          {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦p]] ---> ψ2^[[evar:E↦q]]) abstract (wf_auto2). }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦q]] ---> ψ2^[[evar:E↦p]])
          {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦q]] ---> ψ2^[[evar:E↦p]]) abstract (wf_auto2). }
        }
      }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using i
      3: {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using i
        eapply Framing_right with (ψ := ψ1^[[evar: E ↦ q]]) (wfψ := Hwf2).
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦q]] ↾ Hwf2]}) i
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ2^[[evar:E↦p]] ---> ψ2^[[evar:E↦q]] using i
        1: {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦q]] ↾ Hwf2]}) i eapply pile_trans.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦q]] ↾ Hwf2]}) ?i₂
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe ?i₂ i 2: apply pile.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦q]] ↾ Hwf2]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) try_solve_pile. }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ2^[[evar:E↦p]] ---> ψ2^[[evar:E↦q]] using i
        {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ2^[[evar:E↦p]] ---> ψ2^[[evar:E↦q]] using i
          eapply pf_conj_elim_l_meta in pf₂.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] ---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ2^[[evar:E↦p]] ---> ψ2^[[evar:E↦q]] using i
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦p]] ---> ψ2^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦q]] ---> ψ2^[[evar:E↦p]])
          apply pf₂.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦p]] ---> ψ2^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦q]] ---> ψ2^[[evar:E↦p]])
          {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦p]] ---> ψ2^[[evar:E↦q]]) abstract (wf_auto2). }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦q]] ---> ψ2^[[evar:E↦p]])
          {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ2^[[evar:E↦q]] ---> ψ2^[[evar:E↦p]]) abstract (wf_auto2). }
        }
      }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]])
      {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]])
        abstract (wf_auto2).
      }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]])
      {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]])
        abstract (wf_auto2).
      }
     }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] using i
     4: {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] using i
      apply conj_intro_meta.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] using i
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] using i
      4: {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] using i
        apply Framing_left with (ψ := ψ2^[[evar: E ↦ p]]) (wfψ := Hwf3).
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ2^[[evar:E↦p]] ↾ Hwf3]}) i
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]] using i
        {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ2^[[evar:E↦p]] ↾ Hwf3]}) i eapply pile_trans.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ2^[[evar:E↦p]] ↾ Hwf3]}) ?i₂
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe ?i₂ i 2: apply pile.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ2^[[evar:E↦p]] ↾ Hwf3]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) try_solve_pile. }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]] using i
        {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]] using i
          eapply pf_conj_elim_r_meta in pf₁.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]] using i
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦p]] ---> ψ1^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]])
          apply pf₁.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦p]] ---> ψ1^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]])
          {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦p]] ---> ψ1^[[evar:E↦q]]) abstract (wf_auto2). }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]])
          {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]]) abstract (wf_auto2). }
        }
      }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] using i
      3: {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] using i
        apply Framing_left with (ψ := ψ2^[[evar: E ↦ p]]) (wfψ := Hwf3).
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ2^[[evar:E↦p]] ↾ Hwf3]}) i
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ1^[[evar:E↦p]] ---> ψ1^[[evar:E↦q]] using i
        {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ2^[[evar:E↦p]] ↾ Hwf3]}) i eapply pile_trans.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ2^[[evar:E↦p]] ↾ Hwf3]}) ?i₂
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe ?i₂ i 2: apply pile.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ2^[[evar:E↦p]] ↾ Hwf3]})
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) try_solve_pile. }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ1^[[evar:E↦p]] ---> ψ1^[[evar:E↦q]] using i
        {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ1^[[evar:E↦p]] ---> ψ1^[[evar:E↦q]] using i
          eapply pf_conj_elim_l_meta in pf₁.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] ---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
Γ ⊢i ψ1^[[evar:E↦p]] ---> ψ1^[[evar:E↦q]] using i
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦p]] ---> ψ1^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]])
          apply pf₁.
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦p]] ---> ψ1^[[evar:E↦q]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]])
          {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦p]] ---> ψ1^[[evar:E↦q]]) abstract (wf_auto2). }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]])
          {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] ---> ψ1^[[evar:E↦p]]) abstract (wf_auto2). }
        }
      }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]])
      {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]])
        abstract (wf_auto2).
      }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]])
      {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed
  (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]] --->
   ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]])
        abstract (wf_auto2).
      }
     }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]])
     {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]]) abstract (wf_auto2). }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]])
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]])
     {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦p]]) abstract (wf_auto2). }
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]])
     {
Σ: Signature
Γ: Theory
ψ1, ψ2, p, q: Pattern
E: evar
i: ProofInfo
wfψ1: well_formed ψ1
wfψ2: well_formed ψ2
wfp: well_formed p
wfq: well_formed q
Hwf1: well_formed ψ1^[[evar:E↦p]] = true
HeqHwf1: Hwf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp
  wfψ1
Hwf2: well_formed ψ1^[[evar:E↦q]] = true
HeqHwf2: Hwf2 =
well_formed_free_evar_subst_0 E ψ1 q wfq
  wfψ1
Hwf3: well_formed ψ2^[[evar:E↦p]] = true
HeqHwf3: Hwf3 =
well_formed_free_evar_subst_0 E ψ2 p wfp
  wfψ2
Hwf4: well_formed ψ2^[[evar:E↦q]] = true
HeqHwf4: Hwf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq
  wfψ2
pile: ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]] ↾ Hwf1;
         ψ1^[[evar:E↦q]] ↾ Hwf2;
         ψ2^[[evar:E↦p]] ↾ Hwf3;
         ψ2^[[evar:E↦q]] ↾ Hwf4]}) i
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using i
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using i
well_formed (ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]]) abstract (wf_auto2). }
  Defined.

  Lemma eq_prf_equiv_congruence
    (sz : nat)
    Γ p q
    (wfp : well_formed p)
    (wfq : well_formed q)
    (EvS : EVarSet)
    (SvS : SVarSet)
    E ψ
    (Hsz: size' ψ <= sz)
    (wfψ : well_formed ψ)
    (p_sub_EvS : (free_evars p) ⊆ EvS)
    (q_sub_EvS : (free_evars q) ⊆ EvS)
    (ψ_sub_EvS : (free_evars ψ) ⊆ EvS)
    (E_in_EvS : E ∈ EvS)
    (p_sub_SvS : (free_svars p) ⊆ SvS)
    (q_sub_SvS : (free_svars q) ⊆ SvS)
    (ψ_sub_SvS : (free_svars ψ) ⊆ SvS)
    (exdepth : nat)
    (mudepth : nat)
    (gpi : ProofInfo)
    (pile : ProofInfoLe
     (
       (ExGen := list_to_set (evar_fresh_seq EvS (maximal_exists_depth_of_evar_in_pattern' exdepth E ψ)),
       SVSubst := list_to_set (svar_fresh_seq SvS (maximal_mu_depth_of_evar_in_pattern' mudepth E ψ)),
       KT := if decide (0 = (maximal_mu_depth_of_evar_in_pattern' mudepth E ψ)) is left _ then false else true,
       FP :=  gset_to_coGset (@frames_on_the_way_to_hole' Σ EvS SvS E ψ p q wfψ wfp wfq)
      )
     )
     ( gpi)
    )
    (pf : Γ ⊢i (p <---> q) using ( gpi)) :
        Γ ⊢i (((ψ^[[evar: E ↦ p]]) <---> (ψ^[[evar: E ↦ q]]))) using ( gpi).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
EvS: EVarSet
SvS: SVarSet
E: evar
ψ: Pattern
Hsz: size' ψ ≤ sz
wfψ: well_formed ψ
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
E_in_EvS: E ∈ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
exdepth, mudepth: nat
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS
              E ψ p q wfψ wfp wfq)) gpi
pf: Γ ⊢i p <---> q using gpi
Γ ⊢i ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]] using gpi
  Proof.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
EvS: EVarSet
SvS: SVarSet
E: evar
ψ: Pattern
Hsz: size' ψ ≤ sz
wfψ: well_formed ψ
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
E_in_EvS: E ∈ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
exdepth, mudepth: nat
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS
              E ψ p q wfψ wfp wfq)) gpi
pf: Γ ⊢i p <---> q using gpi
Γ ⊢i ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]] using gpi

    move: ψ wfψ Hsz EvS SvS pile
      p_sub_EvS q_sub_EvS E_in_EvS ψ_sub_EvS
      p_sub_SvS q_sub_SvS ψ_sub_SvS
    .
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                       (maximal_exists_depth_of_evar_in_pattern'
                          exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                         (maximal_mu_depth_of_evar_in_pattern'
                            mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole' EvS SvS E
                    ψ p q wfψ wfp wfq)) gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i ψ^[[evar:E↦p]] <--->
                         ψ^[[evar:E↦q]] using gpi
    induction sz; intros ψ wfψ Hsz EvS SvS pile
      p_sub_EvS q_sub_EvS E_in_EvS ψ_sub_EvS p_sub_SvS q_sub_SvS ψ_sub_SvS.
Σ: Signature
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ 0
EvS: EVarSet
SvS: SVarSet
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS
              E ψ p q wfψ wfp wfq)) gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
Γ ⊢i ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS
              E ψ p q wfψ wfp wfq)) gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
Γ ⊢i ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]] using gpi
    abstract (destruct ψ; simpl in Hsz; lia).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS
              E ψ p q wfψ wfp wfq)) gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
Γ ⊢i ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]] using gpi

    lazymatch type of pile with
    | ProofInfoLe ?st _ => set (i' := st) in *
    end.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
Γ ⊢i ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]] using gpi

    destruct (decide (E ∈ free_evars ψ)) as [HEinψ|HEnotinψ].
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEinψ: E ∈ free_evars ψ
Γ ⊢i ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
Γ ⊢i ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]] using gpi
    2: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
Γ ⊢i ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]] using gpi
      eapply cast_proof'.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
(ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]]) = ?ϕ
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
Γ ⊢i ?ϕ using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
(ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]]) = ?ϕ
        rewrite free_evar_subst_no_occurrence.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
count_evar_occurrences E ψ = 0
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
(ψ <---> ψ^[[evar:E↦q]]) = ?ϕ
        {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
count_evar_occurrences E ψ = 0
          apply count_evar_occurrences_0.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
E ∉ free_evars ψ
          assumption.
        }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
(ψ <---> ψ^[[evar:E↦q]]) = ?ϕ
        rewrite free_evar_subst_no_occurrence.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
count_evar_occurrences E ψ = 0
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
(ψ <---> ψ) = ?ϕ
        {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
count_evar_occurrences E ψ = 0
          apply count_evar_occurrences_0.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
E ∉ free_evars ψ
          assumption.
        }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
(ψ <---> ψ) = ?ϕ
        reflexivity.
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
Γ ⊢i ψ <---> ψ using gpi
      useBasicReasoning.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
Γ ⊢i ψ <---> ψ using BasicReasoning
      apply pf_iff_equiv_refl.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
well_formed ψ
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEnotinψ: E ∉ free_evars ψ
well_formed ψ abstract (wf_auto2). }
    }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed ψ
Hsz: size' ψ ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ψ)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ψ)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ψ)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ψ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ψ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ψ ⊆ SvS
HEinψ: E ∈ free_evars ψ
Γ ⊢i ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]] using gpi

    destruct ψ; simpl in Hsz; simpl.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: evar
wfψ: well_formed (patt_free_evar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_evar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_evar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_evar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_evar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_evar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_evar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_evar x)
Γ
⊢i (if decide (E = x) then p else patt_free_evar x) <--->
   (if decide (E = x) then q else patt_free_evar x)
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: svar
wfψ: well_formed (patt_free_svar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_svar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_svar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_svar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_svar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_svar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_svar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_svar x)
Γ ⊢i patt_free_svar x <---> patt_free_svar x using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_evar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_evar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_evar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_evar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_evar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_evar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_evar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_evar n)
Γ ⊢i patt_bound_evar n <---> patt_bound_evar n
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_svar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_svar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_svar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_svar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_svar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_svar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_svar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_svar n)
Γ ⊢i patt_bound_svar n <---> patt_bound_svar n
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
sigma: symbols
wfψ: well_formed (patt_sym sigma)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_sym sigma))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_sym sigma))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_sym sigma))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_sym sigma) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_sym sigma) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_sym sigma) ⊆ SvS
HEinψ: E ∈ free_evars (patt_sym sigma)
Γ ⊢i patt_sym sigma <---> patt_sym sigma using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
wfψ: well_formed ⊥
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ⊥)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ⊥)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ⊥)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ⊥ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ⊥ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ⊥ ⊆ SvS
HEinψ: E ∈ free_evars ⊥
Γ ⊢i ⊥ <---> ⊥ using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: evar
wfψ: well_formed (patt_free_evar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_evar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_evar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_evar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_evar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_evar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_evar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_evar x)
Γ
⊢i (if decide (E = x) then p else patt_free_evar x) <--->
   (if decide (E = x) then q else patt_free_evar x)
using gpi
      destruct (decide (E = x)).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: evar
wfψ: well_formed (patt_free_evar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_evar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_evar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_evar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_evar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_evar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_evar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_evar x)
e: E = x
Γ ⊢i p <---> q using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: evar
wfψ: well_formed (patt_free_evar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_evar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_evar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_evar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_evar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_evar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_evar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_evar x)
n: E ≠ x
Γ ⊢i patt_free_evar x <---> patt_free_evar x using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: evar
wfψ: well_formed (patt_free_evar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_evar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_evar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_evar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_evar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_evar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_evar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_evar x)
e: E = x
Γ ⊢i p <---> q using gpi
        exact pf.
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: evar
wfψ: well_formed (patt_free_evar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_evar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_evar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_evar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_evar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_evar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_evar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_evar x)
n: E ≠ x
Γ ⊢i patt_free_evar x <---> patt_free_evar x using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: evar
wfψ: well_formed (patt_free_evar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_evar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_evar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_evar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_evar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_evar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_evar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_evar x)
n: E ≠ x
Γ ⊢i patt_free_evar x <---> patt_free_evar x using gpi
        useBasicReasoning.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: evar
wfψ: well_formed (patt_free_evar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_evar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_evar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_evar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_evar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_evar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_evar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_evar x)
n: E ≠ x
Γ ⊢i patt_free_evar x <---> patt_free_evar x
using BasicReasoning
        apply pf_iff_equiv_refl.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: evar
wfψ: well_formed (patt_free_evar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_evar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_evar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_evar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_evar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_evar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_evar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_evar x)
n: E ≠ x
well_formed (patt_free_evar x)
        abstract (wf_auto2).
      }
    }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: svar
wfψ: well_formed (patt_free_svar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_svar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_svar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_svar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_svar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_svar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_svar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_svar x)
Γ ⊢i patt_free_svar x <---> patt_free_svar x using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_evar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_evar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_evar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_evar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_evar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_evar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_evar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_evar n)
Γ ⊢i patt_bound_evar n <---> patt_bound_evar n
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_svar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_svar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_svar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_svar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_svar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_svar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_svar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_svar n)
Γ ⊢i patt_bound_svar n <---> patt_bound_svar n
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
sigma: symbols
wfψ: well_formed (patt_sym sigma)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_sym sigma))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_sym sigma))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_sym sigma))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_sym sigma) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_sym sigma) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_sym sigma) ⊆ SvS
HEinψ: E ∈ free_evars (patt_sym sigma)
Γ ⊢i patt_sym sigma <---> patt_sym sigma using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
wfψ: well_formed ⊥
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ⊥)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ⊥)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ⊥)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ⊥ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ⊥ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ⊥ ⊆ SvS
HEinψ: E ∈ free_evars ⊥
Γ ⊢i ⊥ <---> ⊥ using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: svar
wfψ: well_formed (patt_free_svar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_svar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_svar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_svar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_svar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_svar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_svar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_svar x)
Γ ⊢i patt_free_svar x <---> patt_free_svar x using gpi
      useBasicReasoning.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: svar
wfψ: well_formed (patt_free_svar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_svar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_svar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_svar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_svar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_svar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_svar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_svar x)
Γ ⊢i patt_free_svar x <---> patt_free_svar x
using BasicReasoning
      apply pf_iff_equiv_refl.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
x: svar
wfψ: well_formed (patt_free_svar x)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_free_svar x))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_free_svar x))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_free_svar x))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_free_svar x) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_free_svar x) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_free_svar x) ⊆ SvS
HEinψ: E ∈ free_evars (patt_free_svar x)
well_formed (patt_free_svar x)
      abstract (wf_auto2).
    }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_evar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_evar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_evar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_evar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_evar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_evar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_evar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_evar n)
Γ ⊢i patt_bound_evar n <---> patt_bound_evar n
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_svar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_svar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_svar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_svar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_svar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_svar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_svar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_svar n)
Γ ⊢i patt_bound_svar n <---> patt_bound_svar n
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
sigma: symbols
wfψ: well_formed (patt_sym sigma)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_sym sigma))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_sym sigma))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_sym sigma))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_sym sigma) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_sym sigma) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_sym sigma) ⊆ SvS
HEinψ: E ∈ free_evars (patt_sym sigma)
Γ ⊢i patt_sym sigma <---> patt_sym sigma using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
wfψ: well_formed ⊥
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ⊥)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ⊥)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ⊥)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ⊥ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ⊥ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ⊥ ⊆ SvS
HEinψ: E ∈ free_evars ⊥
Γ ⊢i ⊥ <---> ⊥ using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_evar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_evar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_evar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_evar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_evar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_evar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_evar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_evar n)
Γ ⊢i patt_bound_evar n <---> patt_bound_evar n
using gpi
      useBasicReasoning.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_evar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_evar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_evar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_evar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_evar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_evar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_evar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_evar n)
Γ ⊢i patt_bound_evar n <---> patt_bound_evar n
using BasicReasoning
      apply pf_iff_equiv_refl.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_evar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_evar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_evar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_evar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_evar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_evar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_evar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_evar n)
well_formed (patt_bound_evar n)
      abstract (wf_auto2).
    }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_svar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_svar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_svar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_svar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_svar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_svar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_svar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_svar n)
Γ ⊢i patt_bound_svar n <---> patt_bound_svar n
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
sigma: symbols
wfψ: well_formed (patt_sym sigma)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_sym sigma))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_sym sigma))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_sym sigma))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_sym sigma) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_sym sigma) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_sym sigma) ⊆ SvS
HEinψ: E ∈ free_evars (patt_sym sigma)
Γ ⊢i patt_sym sigma <---> patt_sym sigma using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
wfψ: well_formed ⊥
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ⊥)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ⊥)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ⊥)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ⊥ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ⊥ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ⊥ ⊆ SvS
HEinψ: E ∈ free_evars ⊥
Γ ⊢i ⊥ <---> ⊥ using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_svar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_svar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_svar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_svar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_svar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_svar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_svar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_svar n)
Γ ⊢i patt_bound_svar n <---> patt_bound_svar n
using gpi
      useBasicReasoning.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_svar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_svar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_svar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_svar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_svar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_svar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_svar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_svar n)
Γ ⊢i patt_bound_svar n <---> patt_bound_svar n
using BasicReasoning
      apply pf_iff_equiv_refl.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
n: db_index
wfψ: well_formed (patt_bound_svar n)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_bound_svar n))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_bound_svar n))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_bound_svar n))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_bound_svar n) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_bound_svar n) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_bound_svar n) ⊆ SvS
HEinψ: E ∈ free_evars (patt_bound_svar n)
well_formed (patt_bound_svar n)
      abstract (wf_auto2).
    }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
sigma: symbols
wfψ: well_formed (patt_sym sigma)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_sym sigma))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_sym sigma))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_sym sigma))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_sym sigma) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_sym sigma) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_sym sigma) ⊆ SvS
HEinψ: E ∈ free_evars (patt_sym sigma)
Γ ⊢i patt_sym sigma <---> patt_sym sigma using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
wfψ: well_formed ⊥
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ⊥)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ⊥)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ⊥)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ⊥ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ⊥ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ⊥ ⊆ SvS
HEinψ: E ∈ free_evars ⊥
Γ ⊢i ⊥ <---> ⊥ using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
sigma: symbols
wfψ: well_formed (patt_sym sigma)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_sym sigma))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_sym sigma))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_sym sigma))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_sym sigma) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_sym sigma) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_sym sigma) ⊆ SvS
HEinψ: E ∈ free_evars (patt_sym sigma)
Γ ⊢i patt_sym sigma <---> patt_sym sigma using gpi
      useBasicReasoning.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
sigma: symbols
wfψ: well_formed (patt_sym sigma)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_sym sigma))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_sym sigma))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_sym sigma))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_sym sigma) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_sym sigma) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_sym sigma) ⊆ SvS
HEinψ: E ∈ free_evars (patt_sym sigma)
Γ ⊢i patt_sym sigma <---> patt_sym sigma
using BasicReasoning
      apply pf_iff_equiv_refl.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
sigma: symbols
wfψ: well_formed (patt_sym sigma)
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (patt_sym sigma))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (patt_sym sigma))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (patt_sym sigma))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (patt_sym sigma) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (patt_sym sigma) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (patt_sym sigma) ⊆ SvS
HEinψ: E ∈ free_evars (patt_sym sigma)
well_formed (patt_sym sigma)
      abstract (wf_auto2).
    }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
wfψ: well_formed ⊥
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ⊥)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ⊥)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ⊥)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ⊥ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ⊥ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ⊥ ⊆ SvS
HEinψ: E ∈ free_evars ⊥
Γ ⊢i ⊥ <---> ⊥ using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      assert (wfψ1 : well_formed ψ1 = true).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
well_formed ψ1 = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
well_formed ψ1 = true clear -wfψ.
Σ: Signature
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
well_formed ψ1 = true abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      assert (size' ψ1 <= sz) by abstract(lia).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      assert (wfψ2 : well_formed ψ2 = true).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
well_formed ψ2 = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
well_formed ψ2 = true clear -wfψ.
Σ: Signature
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
well_formed ψ2 = true abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      assert (size' ψ2 <= sz) by abstract(lia).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      
      pose proof (pf₁ := (IHsz ψ1) ltac:(assumption) ltac:(assumption)).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ∀ (EvS : EVarSet) (SvS : SVarSet),
  ProofInfoLe
    (ExGen := list_to_set
                (evar_fresh_seq EvS
                   (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ1)),
     SVSubst := list_to_set
                  (svar_fresh_seq SvS
                     (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
     KT := if
            decide
              (0 =
               maximal_mu_depth_of_evar_in_pattern'
                 mudepth E ψ1)
           then false
           else true,
     FP := gset_to_coGset
             (frames_on_the_way_to_hole' EvS SvS
                E ψ1 p q wfψ1 wfp wfq)) gpi
  → free_evars p ⊆ EvS
    → free_evars q ⊆ EvS
      → E ∈ EvS
        → free_evars ψ1 ⊆ EvS
          → free_svars p ⊆ SvS
            → free_svars q ⊆ SvS
              → free_svars ψ1 ⊆ SvS
                → Γ
                  ⊢i ψ1^[[evar:E↦p]] <--->
                     ψ1^[[evar:E↦q]] using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      specialize (pf₁ EvS SvS).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      feed specialize pf₁.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E ψ1 p
              q wfψ1 wfp wfq)) gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E ψ1 p
              q wfψ1 wfp wfq)) gpi
        abstract (
          eapply pile_trans;[|apply pile];
          subst i';
        apply pile_evs_svs_kt;
        [
        (
          simpl; clear; pose proof (Htmp := evar_fresh_seq_max); set_solver
        )|
        (
          simpl; clear; pose proof (Htmp := svar_fresh_seq_max); set_solver
        )|
        (
          simpl; clear pf₁;
          repeat case_match; simpl; try reflexivity; lia
        )|(simpl; apply frames_on_the_way_to_hole'_app_1)
        ]).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS exact p_sub_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS exact q_sub_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
E ∈ EvS exact E_in_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS abstract(
          simpl in ψ_sub_EvS;
          clear -ψ_sub_EvS;
          set_solver
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS exact p_sub_SvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS exact q_sub_SvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS abstract(
          simpl in ψ_sub_SvS;
          clear -ψ_sub_SvS;
          set_solver
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      pose proof (pf₂ := (IHsz ψ2) ltac:(assumption) ltac:(assumption)).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ∀ (EvS : EVarSet) (SvS : SVarSet),
  ProofInfoLe
    (ExGen := list_to_set
                (evar_fresh_seq EvS
                   (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ2)),
     SVSubst := list_to_set
                  (svar_fresh_seq SvS
                     (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
     KT := if
            decide
              (0 =
               maximal_mu_depth_of_evar_in_pattern'
                 mudepth E ψ2)
           then false
           else true,
     FP := gset_to_coGset
             (frames_on_the_way_to_hole' EvS SvS
                E ψ2 p q wfψ2 wfp wfq)) gpi
  → free_evars p ⊆ EvS
    → free_evars q ⊆ EvS
      → E ∈ EvS
        → free_evars ψ2 ⊆ EvS
          → free_svars p ⊆ SvS
            → free_svars q ⊆ SvS
              → free_svars ψ2 ⊆ SvS
                → Γ
                  ⊢i ψ2^[[evar:E↦p]] <--->
                     ψ2^[[evar:E↦q]] using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      specialize (pf₂ EvS SvS).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      feed specialize pf₂.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E ψ2 p
              q wfψ2 wfp wfq)) gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E ψ2 p
              q wfψ2 wfp wfq)) gpi
        abstract (subst i';
          eapply pile_trans;[|(apply pile)];
          simpl;
          apply pile_evs_svs_kt;
          [(simpl; rewrite Nat.max_comm; pose proof (Htmp := evar_fresh_seq_max); set_solver)
          |(rewrite Nat.max_comm; pose proof (Htmp := svar_fresh_seq_max); set_solver)
          |(clear pf₂; repeat case_match; simpl; try reflexivity; try lia)
          |(apply frames_on_the_way_to_hole'_app_2)
          ]
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS exact p_sub_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS exact q_sub_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
E ∈ EvS exact E_in_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
        abstract (
          simpl in ψ_sub_EvS;
          clear -ψ_sub_EvS;
          set_solver
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS exact p_sub_SvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS exact q_sub_SvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS 
        abstract (
          simpl in ψ_sub_SvS;
          clear -ψ_sub_SvS;
          set_solver
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i ψ1^[[evar:E↦p]] $ ψ2^[[evar:E↦p]] <--->
   ψ1^[[evar:E↦q]] $ ψ2^[[evar:E↦q]] using gpi

      unshelve (eapply helper_app_lemma); try assumption.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]]
           ↾ well_formed_free_evar_subst_0 E ψ1 p wfp
               wfψ1;
         ψ1^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ1 q wfq
             wfψ1;
         ψ2^[[evar:E↦p]]
         ↾ well_formed_free_evar_subst_0 E ψ2 p wfp
             wfψ2;
         ψ2^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ2 q wfq
             wfψ2]}) gpi
      eapply pile_trans;[|apply pile].
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]]
           ↾ well_formed_free_evar_subst_0 E ψ1 p wfp
               wfψ1;
         ψ1^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ1 q wfq
             wfψ1;
         ψ2^[[evar:E↦p]]
         ↾ well_formed_free_evar_subst_0 E ψ2 p wfp
             wfψ2;
         ψ2^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ2 q wfq
             wfψ2]}) i'
      unfold i'.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := {[ψ1^[[evar:E↦p]]
           ↾ well_formed_free_evar_subst_0 E ψ1 p wfp
               wfψ1;
         ψ1^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ1 q wfq
             wfψ1;
         ψ2^[[evar:E↦p]]
         ↾ well_formed_free_evar_subst_0 E ψ2 p wfp
             wfψ2;
         ψ2^[[evar:E↦q]]
         ↾ well_formed_free_evar_subst_0 E ψ2 q wfq
             wfψ2]})
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E (ψ1 $ ψ2))),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E (ψ1 $ ψ2))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E (ψ1 $ ψ2))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              (ψ1 $ ψ2) p q wfψ wfp wfq))
      apply pile_evs_svs_kt.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
∅
⊆ list_to_set
    (evar_fresh_seq EvS
       (maximal_exists_depth_of_evar_in_pattern'
          exdepth E (ψ1 $ ψ2)))
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
∅
⊆ list_to_set
    (svar_fresh_seq SvS
       (maximal_mu_depth_of_evar_in_pattern' mudepth E
          (ψ1 $ ψ2)))
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
false ==>
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ψ1 $ ψ2))
 then false
 else true)
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
{[ψ1^[[evar:E↦p]]
  ↾ well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1;
ψ1^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1;
ψ2^[[evar:E↦p]]
↾ well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2;
ψ2^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2]}
⊆ gset_to_coGset
    (frames_on_the_way_to_hole' EvS SvS E 
       (ψ1 $ ψ2) p q wfψ wfp wfq)
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
∅
⊆ list_to_set
    (evar_fresh_seq EvS
       (maximal_exists_depth_of_evar_in_pattern'
          exdepth E (ψ1 $ ψ2))) clear; set_solver. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
∅
⊆ list_to_set
    (svar_fresh_seq SvS
       (maximal_mu_depth_of_evar_in_pattern' mudepth E
          (ψ1 $ ψ2)))
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
false ==>
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ψ1 $ ψ2))
 then false
 else true)
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
{[ψ1^[[evar:E↦p]]
  ↾ well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1;
ψ1^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1;
ψ2^[[evar:E↦p]]
↾ well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2;
ψ2^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2]}
⊆ gset_to_coGset
    (frames_on_the_way_to_hole' EvS SvS E 
       (ψ1 $ ψ2) p q wfψ wfp wfq)
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
∅
⊆ list_to_set
    (svar_fresh_seq SvS
       (maximal_mu_depth_of_evar_in_pattern' mudepth E
          (ψ1 $ ψ2))) clear; set_solver. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
false ==>
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ψ1 $ ψ2))
 then false
 else true)
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
{[ψ1^[[evar:E↦p]]
  ↾ well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1;
ψ1^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1;
ψ2^[[evar:E↦p]]
↾ well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2;
ψ2^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2]}
⊆ gset_to_coGset
    (frames_on_the_way_to_hole' EvS SvS E 
       (ψ1 $ ψ2) p q wfψ wfp wfq)
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
false ==>
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ψ1 $ ψ2))
 then false
 else true) reflexivity. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
{[ψ1^[[evar:E↦p]]
  ↾ well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1;
ψ1^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1;
ψ2^[[evar:E↦p]]
↾ well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2;
ψ2^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2]}
⊆ gset_to_coGset
    (frames_on_the_way_to_hole' EvS SvS E (ψ1 $ ψ2) p
       q wfψ wfp wfq)
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
{[ψ1^[[evar:E↦p]]
  ↾ well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1;
ψ1^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1;
ψ2^[[evar:E↦p]]
↾ well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2;
ψ2^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2]}
⊆ gset_to_coGset
    (frames_on_the_way_to_hole' EvS SvS E (ψ1 $ ψ2) p
       q wfψ wfp wfq)
        simp frames_on_the_way_to_hole'.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
{[ψ1^[[evar:E↦p]]
  ↾ well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1;
ψ1^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1;
ψ2^[[evar:E↦p]]
↾ well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2;
ψ2^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2]}
⊆ gset_to_coGset
    ({[ψ1^[[evar:E↦p]]
       ↾ frames_on_the_way_to_hole'_obligation_1 E ψ1
           ψ2 p wfψ wfp;
     ψ2^[[evar:E↦p]]
     ↾ frames_on_the_way_to_hole'_obligation_2 E ψ1 ψ2
         p wfψ wfp;
     ψ1^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_3 E ψ1 ψ2
         q wfψ wfq;
     ψ2^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_4 E ψ1 ψ2
         q wfψ wfq]}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        (* This should be automatable! *)
        remember (well_formed_free_evar_subst_0 E _ _ wfp wfψ1) as wf1.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
wf1: well_formed ψ1^[[evar:E↦p]] = true
Heqwf1: wf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1
{[ψ1^[[evar:E↦p]] ↾ wf1;
ψ1^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1;
ψ2^[[evar:E↦p]]
↾ well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2;
ψ2^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2]}
⊆ gset_to_coGset
    ({[ψ1^[[evar:E↦p]]
       ↾ frames_on_the_way_to_hole'_obligation_1 E ψ1
           ψ2 p wfψ wfp;
     ψ2^[[evar:E↦p]]
     ↾ frames_on_the_way_to_hole'_obligation_2 E ψ1 ψ2
         p wfψ wfp;
     ψ1^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_3 E ψ1 ψ2
         q wfψ wfq;
     ψ2^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_4 E ψ1 ψ2
         q wfψ wfq]}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (well_formed_free_evar_subst_0 E _ _ wfp wfψ2) as wf2.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
wf1: well_formed ψ1^[[evar:E↦p]] = true
Heqwf1: wf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1
wf2: well_formed ψ2^[[evar:E↦p]] = true
Heqwf2: wf2 =
well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2
{[ψ1^[[evar:E↦p]] ↾ wf1;
ψ1^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1;
ψ2^[[evar:E↦p]] ↾ wf2;
ψ2^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2]}
⊆ gset_to_coGset
    ({[ψ1^[[evar:E↦p]]
       ↾ frames_on_the_way_to_hole'_obligation_1 E ψ1
           ψ2 p wfψ wfp;
     ψ2^[[evar:E↦p]]
     ↾ frames_on_the_way_to_hole'_obligation_2 E ψ1 ψ2
         p wfψ wfp;
     ψ1^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_3 E ψ1 ψ2
         q wfψ wfq;
     ψ2^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_4 E ψ1 ψ2
         q wfψ wfq]}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (well_formed_free_evar_subst_0 E _ _ wfq wfψ1) as wf3.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
wf1: well_formed ψ1^[[evar:E↦p]] = true
Heqwf1: wf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1
wf2: well_formed ψ2^[[evar:E↦p]] = true
Heqwf2: wf2 =
well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2
wf3: well_formed ψ1^[[evar:E↦q]] = true
Heqwf3: wf3 =
well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1
{[ψ1^[[evar:E↦p]] ↾ wf1; ψ1^[[evar:E↦q]] ↾ wf3;
ψ2^[[evar:E↦p]] ↾ wf2;
ψ2^[[evar:E↦q]]
↾ well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2]}
⊆ gset_to_coGset
    ({[ψ1^[[evar:E↦p]]
       ↾ frames_on_the_way_to_hole'_obligation_1 E ψ1
           ψ2 p wfψ wfp;
     ψ2^[[evar:E↦p]]
     ↾ frames_on_the_way_to_hole'_obligation_2 E ψ1 ψ2
         p wfψ wfp;
     ψ1^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_3 E ψ1 ψ2
         q wfψ wfq;
     ψ2^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_4 E ψ1 ψ2
         q wfψ wfq]}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (well_formed_free_evar_subst_0 E _ _ wfq wfψ2) as wf4.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
wf1: well_formed ψ1^[[evar:E↦p]] = true
Heqwf1: wf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1
wf2: well_formed ψ2^[[evar:E↦p]] = true
Heqwf2: wf2 =
well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2
wf3: well_formed ψ1^[[evar:E↦q]] = true
Heqwf3: wf3 =
well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1
wf4: well_formed ψ2^[[evar:E↦q]] = true
Heqwf4: wf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2
{[ψ1^[[evar:E↦p]] ↾ wf1; ψ1^[[evar:E↦q]] ↾ wf3;
ψ2^[[evar:E↦p]] ↾ wf2; ψ2^[[evar:E↦q]] ↾ wf4]}
⊆ gset_to_coGset
    ({[ψ1^[[evar:E↦p]]
       ↾ frames_on_the_way_to_hole'_obligation_1 E ψ1
           ψ2 p wfψ wfp;
     ψ2^[[evar:E↦p]]
     ↾ frames_on_the_way_to_hole'_obligation_2 E ψ1 ψ2
         p wfψ wfp;
     ψ1^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_3 E ψ1 ψ2
         q wfψ wfq;
     ψ2^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_4 E ψ1 ψ2
         q wfψ wfq]}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (frames_on_the_way_to_hole' EvS SvS E _ _ _
        (frames_on_the_way_to_hole'_obligation_5 _ _ wfψ)
        (frames_on_the_way_to_hole'_obligation_6 _ wfp)
        (frames_on_the_way_to_hole'_obligation_7 _ wfq) ∪
      frames_on_the_way_to_hole' EvS SvS E _ _ _
        (frames_on_the_way_to_hole'_obligation_9 _ _ wfψ)
        (frames_on_the_way_to_hole'_obligation_10 _ wfp)
        (frames_on_the_way_to_hole'_obligation_11 _ wfq))
        as rest.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
wf1: well_formed ψ1^[[evar:E↦p]] = true
Heqwf1: wf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1
wf2: well_formed ψ2^[[evar:E↦p]] = true
Heqwf2: wf2 =
well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2
wf3: well_formed ψ1^[[evar:E↦q]] = true
Heqwf3: wf3 =
well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1
wf4: well_formed ψ2^[[evar:E↦q]] = true
Heqwf4: wf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2
rest: gset wfPattern
Heqrest: rest =
frames_on_the_way_to_hole' EvS SvS E ψ1 p q
  (frames_on_the_way_to_hole'_obligation_5
     ψ1 ψ2 wfψ)
  (frames_on_the_way_to_hole'_obligation_6 p
     wfp)
  (frames_on_the_way_to_hole'_obligation_7 q
     wfq)
∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p
    q
    (frames_on_the_way_to_hole'_obligation_9
       ψ1 ψ2 wfψ)
    (frames_on_the_way_to_hole'_obligation_10
       p wfp)
    (frames_on_the_way_to_hole'_obligation_11
       q wfq)
{[ψ1^[[evar:E↦p]] ↾ wf1; ψ1^[[evar:E↦q]] ↾ wf3;
ψ2^[[evar:E↦p]] ↾ wf2; ψ2^[[evar:E↦q]] ↾ wf4]}
⊆ gset_to_coGset
    ({[ψ1^[[evar:E↦p]]
       ↾ frames_on_the_way_to_hole'_obligation_1 E ψ1
           ψ2 p wfψ wfp;
     ψ2^[[evar:E↦p]]
     ↾ frames_on_the_way_to_hole'_obligation_2 E ψ1 ψ2
         p wfψ wfp;
     ψ1^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_3 E ψ1 ψ2
         q wfψ wfq;
     ψ2^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_4 E ψ1 ψ2
         q wfψ wfq]}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (frames_on_the_way_to_hole'_obligation_1 E _ _ _ wfψ wfp) as wf1'.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
wf1: well_formed ψ1^[[evar:E↦p]] = true
Heqwf1: wf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1
wf2: well_formed ψ2^[[evar:E↦p]] = true
Heqwf2: wf2 =
well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2
wf3: well_formed ψ1^[[evar:E↦q]] = true
Heqwf3: wf3 =
well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1
wf4: well_formed ψ2^[[evar:E↦q]] = true
Heqwf4: wf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2
rest: gset wfPattern
Heqrest: rest =
frames_on_the_way_to_hole' EvS SvS E ψ1 p q
  (frames_on_the_way_to_hole'_obligation_5
     ψ1 ψ2 wfψ)
  (frames_on_the_way_to_hole'_obligation_6 p
     wfp)
  (frames_on_the_way_to_hole'_obligation_7 q
     wfq)
∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p
    q
    (frames_on_the_way_to_hole'_obligation_9
       ψ1 ψ2 wfψ)
    (frames_on_the_way_to_hole'_obligation_10
       p wfp)
    (frames_on_the_way_to_hole'_obligation_11
       q wfq)
wf1': well_formed ψ1^[[evar:E↦p]] = true
Heqwf1': wf1' =
frames_on_the_way_to_hole'_obligation_1 E ψ1
  ψ2 p wfψ wfp
{[ψ1^[[evar:E↦p]] ↾ wf1; ψ1^[[evar:E↦q]] ↾ wf3;
ψ2^[[evar:E↦p]] ↾ wf2; ψ2^[[evar:E↦q]] ↾ wf4]}
⊆ gset_to_coGset
    ({[ψ1^[[evar:E↦p]] ↾ wf1';
     ψ2^[[evar:E↦p]]
     ↾ frames_on_the_way_to_hole'_obligation_2 E ψ1 ψ2
         p wfψ wfp;
     ψ1^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_3 E ψ1 ψ2
         q wfψ wfq;
     ψ2^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_4 E ψ1 ψ2
         q wfψ wfq]}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (frames_on_the_way_to_hole'_obligation_2 E _ _ _ wfψ wfp) as wf2'.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
wf1: well_formed ψ1^[[evar:E↦p]] = true
Heqwf1: wf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1
wf2: well_formed ψ2^[[evar:E↦p]] = true
Heqwf2: wf2 =
well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2
wf3: well_formed ψ1^[[evar:E↦q]] = true
Heqwf3: wf3 =
well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1
wf4: well_formed ψ2^[[evar:E↦q]] = true
Heqwf4: wf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2
rest: gset wfPattern
Heqrest: rest =
frames_on_the_way_to_hole' EvS SvS E ψ1 p q
  (frames_on_the_way_to_hole'_obligation_5
     ψ1 ψ2 wfψ)
  (frames_on_the_way_to_hole'_obligation_6 p
     wfp)
  (frames_on_the_way_to_hole'_obligation_7 q
     wfq)
∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p
    q
    (frames_on_the_way_to_hole'_obligation_9
       ψ1 ψ2 wfψ)
    (frames_on_the_way_to_hole'_obligation_10
       p wfp)
    (frames_on_the_way_to_hole'_obligation_11
       q wfq)
wf1': well_formed ψ1^[[evar:E↦p]] = true
Heqwf1': wf1' =
frames_on_the_way_to_hole'_obligation_1 E ψ1
  ψ2 p wfψ wfp
wf2': well_formed ψ2^[[evar:E↦p]] = true
Heqwf2': wf2' =
frames_on_the_way_to_hole'_obligation_2 E ψ1
  ψ2 p wfψ wfp
{[ψ1^[[evar:E↦p]] ↾ wf1; ψ1^[[evar:E↦q]] ↾ wf3;
ψ2^[[evar:E↦p]] ↾ wf2; ψ2^[[evar:E↦q]] ↾ wf4]}
⊆ gset_to_coGset
    ({[ψ1^[[evar:E↦p]] ↾ wf1'; ψ2^[[evar:E↦p]] ↾ wf2';
     ψ1^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_3 E ψ1 ψ2
         q wfψ wfq;
     ψ2^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_4 E ψ1 ψ2
         q wfψ wfq]}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (frames_on_the_way_to_hole'_obligation_3 E _ _ _ wfψ wfq) as wf3'.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
wf1: well_formed ψ1^[[evar:E↦p]] = true
Heqwf1: wf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1
wf2: well_formed ψ2^[[evar:E↦p]] = true
Heqwf2: wf2 =
well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2
wf3: well_formed ψ1^[[evar:E↦q]] = true
Heqwf3: wf3 =
well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1
wf4: well_formed ψ2^[[evar:E↦q]] = true
Heqwf4: wf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2
rest: gset wfPattern
Heqrest: rest =
frames_on_the_way_to_hole' EvS SvS E ψ1 p q
  (frames_on_the_way_to_hole'_obligation_5
     ψ1 ψ2 wfψ)
  (frames_on_the_way_to_hole'_obligation_6 p
     wfp)
  (frames_on_the_way_to_hole'_obligation_7 q
     wfq)
∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p
    q
    (frames_on_the_way_to_hole'_obligation_9
       ψ1 ψ2 wfψ)
    (frames_on_the_way_to_hole'_obligation_10
       p wfp)
    (frames_on_the_way_to_hole'_obligation_11
       q wfq)
wf1': well_formed ψ1^[[evar:E↦p]] = true
Heqwf1': wf1' =
frames_on_the_way_to_hole'_obligation_1 E ψ1
  ψ2 p wfψ wfp
wf2': well_formed ψ2^[[evar:E↦p]] = true
Heqwf2': wf2' =
frames_on_the_way_to_hole'_obligation_2 E ψ1
  ψ2 p wfψ wfp
wf3': well_formed ψ1^[[evar:E↦q]] = true
Heqwf3': wf3' =
frames_on_the_way_to_hole'_obligation_3 E ψ1
  ψ2 q wfψ wfq
{[ψ1^[[evar:E↦p]] ↾ wf1; ψ1^[[evar:E↦q]] ↾ wf3;
ψ2^[[evar:E↦p]] ↾ wf2; ψ2^[[evar:E↦q]] ↾ wf4]}
⊆ gset_to_coGset
    ({[ψ1^[[evar:E↦p]] ↾ wf1'; ψ2^[[evar:E↦p]] ↾ wf2';
     ψ1^[[evar:E↦q]] ↾ wf3';
     ψ2^[[evar:E↦q]]
     ↾ frames_on_the_way_to_hole'_obligation_4 E ψ1 ψ2
         q wfψ wfq]}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (frames_on_the_way_to_hole'_obligation_4 E _ _ _ wfψ wfq) as wf4'.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 $ ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 $ ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 $ ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 $ ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 $ ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 $ ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 $ ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
wf1: well_formed ψ1^[[evar:E↦p]] = true
Heqwf1: wf1 =
well_formed_free_evar_subst_0 E ψ1 p wfp wfψ1
wf2: well_formed ψ2^[[evar:E↦p]] = true
Heqwf2: wf2 =
well_formed_free_evar_subst_0 E ψ2 p wfp wfψ2
wf3: well_formed ψ1^[[evar:E↦q]] = true
Heqwf3: wf3 =
well_formed_free_evar_subst_0 E ψ1 q wfq wfψ1
wf4: well_formed ψ2^[[evar:E↦q]] = true
Heqwf4: wf4 =
well_formed_free_evar_subst_0 E ψ2 q wfq wfψ2
rest: gset wfPattern
Heqrest: rest =
frames_on_the_way_to_hole' EvS SvS E ψ1 p q
  (frames_on_the_way_to_hole'_obligation_5
     ψ1 ψ2 wfψ)
  (frames_on_the_way_to_hole'_obligation_6 p
     wfp)
  (frames_on_the_way_to_hole'_obligation_7 q
     wfq)
∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p
    q
    (frames_on_the_way_to_hole'_obligation_9
       ψ1 ψ2 wfψ)
    (frames_on_the_way_to_hole'_obligation_10
       p wfp)
    (frames_on_the_way_to_hole'_obligation_11
       q wfq)
wf1': well_formed ψ1^[[evar:E↦p]] = true
Heqwf1': wf1' =
frames_on_the_way_to_hole'_obligation_1 E ψ1
  ψ2 p wfψ wfp
wf2': well_formed ψ2^[[evar:E↦p]] = true
Heqwf2': wf2' =
frames_on_the_way_to_hole'_obligation_2 E ψ1
  ψ2 p wfψ wfp
wf3': well_formed ψ1^[[evar:E↦q]] = true
Heqwf3': wf3' =
frames_on_the_way_to_hole'_obligation_3 E ψ1
  ψ2 q wfψ wfq
wf4': well_formed ψ2^[[evar:E↦q]] = true
Heqwf4': wf4' =
frames_on_the_way_to_hole'_obligation_4 E ψ1
  ψ2 q wfψ wfq
{[ψ1^[[evar:E↦p]] ↾ wf1; ψ1^[[evar:E↦q]] ↾ wf3;
ψ2^[[evar:E↦p]] ↾ wf2; ψ2^[[evar:E↦q]] ↾ wf4]}
⊆ gset_to_coGset
    ({[ψ1^[[evar:E↦p]] ↾ wf1'; ψ2^[[evar:E↦p]] ↾ wf2';
     ψ1^[[evar:E↦q]] ↾ wf3'; ψ2^[[evar:E↦q]] ↾ wf4']}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        clear.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
wf1: well_formed ψ1^[[evar:E↦p]] = true
wf2: well_formed ψ2^[[evar:E↦p]] = true
wf3: well_formed ψ1^[[evar:E↦q]] = true
wf4: well_formed ψ2^[[evar:E↦q]] = true
wf1': well_formed ψ1^[[evar:E↦p]] = true
wf2': well_formed ψ2^[[evar:E↦p]] = true
wf3': well_formed ψ1^[[evar:E↦q]] = true
wf4': well_formed ψ2^[[evar:E↦q]] = true
{[ψ1^[[evar:E↦p]] ↾ wf1; ψ1^[[evar:E↦q]] ↾ wf3;
ψ2^[[evar:E↦p]] ↾ wf2; ψ2^[[evar:E↦q]] ↾ wf4]}
⊆ gset_to_coGset
    ({[ψ1^[[evar:E↦p]] ↾ wf1'; ψ2^[[evar:E↦p]] ↾ wf2';
     ψ1^[[evar:E↦q]] ↾ wf3'; ψ2^[[evar:E↦q]] ↾ wf4']}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (ψ1^[[evar: E ↦ p]]) as A.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
HeqA: A = ψ1^[[evar:E↦p]]
wf1: well_formed A = true
wf2: well_formed ψ2^[[evar:E↦p]] = true
wf3: well_formed ψ1^[[evar:E↦q]] = true
wf4: well_formed ψ2^[[evar:E↦q]] = true
wf1': well_formed A = true
wf2': well_formed ψ2^[[evar:E↦p]] = true
wf3': well_formed ψ1^[[evar:E↦q]] = true
wf4': well_formed ψ2^[[evar:E↦q]] = true
{[A ↾ wf1; ψ1^[[evar:E↦q]] ↾ wf3;
ψ2^[[evar:E↦p]] ↾ wf2; ψ2^[[evar:E↦q]] ↾ wf4]}
⊆ gset_to_coGset
    ({[A ↾ wf1'; ψ2^[[evar:E↦p]] ↾ wf2';
     ψ1^[[evar:E↦q]] ↾ wf3'; ψ2^[[evar:E↦q]] ↾ wf4']}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (ψ1^[[evar: E ↦ q]]) as B.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
HeqA: A = ψ1^[[evar:E↦p]]
wf1: well_formed A = true
wf2: well_formed ψ2^[[evar:E↦p]] = true
B: Pattern
HeqB: B = ψ1^[[evar:E↦q]]
wf3: well_formed B = true
wf4: well_formed ψ2^[[evar:E↦q]] = true
wf1': well_formed A = true
wf2': well_formed ψ2^[[evar:E↦p]] = true
wf3': well_formed B = true
wf4': well_formed ψ2^[[evar:E↦q]] = true
{[A ↾ wf1; B ↾ wf3; ψ2^[[evar:E↦p]] ↾ wf2;
ψ2^[[evar:E↦q]] ↾ wf4]}
⊆ gset_to_coGset
    ({[A ↾ wf1'; ψ2^[[evar:E↦p]] ↾ wf2'; B ↾ wf3';
     ψ2^[[evar:E↦q]] ↾ wf4']}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (ψ2^[[evar: E ↦ p]]) as C.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
HeqA: A = ψ1^[[evar:E↦p]]
wf1: well_formed A = true
C: Pattern
HeqC: C = ψ2^[[evar:E↦p]]
wf2: well_formed C = true
B: Pattern
HeqB: B = ψ1^[[evar:E↦q]]
wf3: well_formed B = true
wf4: well_formed ψ2^[[evar:E↦q]] = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed ψ2^[[evar:E↦q]] = true
{[A ↾ wf1; B ↾ wf3; C ↾ wf2; ψ2^[[evar:E↦q]] ↾ wf4]}
⊆ gset_to_coGset
    ({[A ↾ wf1'; C ↾ wf2'; B ↾ wf3';
     ψ2^[[evar:E↦q]] ↾ wf4']}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        remember (ψ2^[[evar: E ↦ q]]) as D.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
HeqA: A = ψ1^[[evar:E↦p]]
wf1: well_formed A = true
C: Pattern
HeqC: C = ψ2^[[evar:E↦p]]
wf2: well_formed C = true
B: Pattern
HeqB: B = ψ1^[[evar:E↦q]]
wf3: well_formed B = true
D: Pattern
HeqD: D = ψ2^[[evar:E↦q]]
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
{[A ↾ wf1; B ↾ wf3; C ↾ wf2; D ↾ wf4]}
⊆ gset_to_coGset
    ({[A ↾ wf1'; C ↾ wf2'; B ↾ wf3'; D ↾ wf4']}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        clear.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
{[A ↾ wf1; B ↾ wf3; C ↾ wf2; D ↾ wf4]}
⊆ gset_to_coGset
    ({[A ↾ wf1'; C ↾ wf2'; B ↾ wf3'; D ↾ wf4']}
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
         (frames_on_the_way_to_hole'_obligation_5 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_6 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_7 q
            wfq)
     ∪ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
         (frames_on_the_way_to_hole'_obligation_9 ψ1
            ψ2 wfψ)
         (frames_on_the_way_to_hole'_obligation_10 p
            wfp)
         (frames_on_the_way_to_hole'_obligation_11 q
            wfq))
        set_unfold.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
∀ x : wfPattern,
  ((x = A ↾ wf1 ∨ x = B ↾ wf3) ∨ x = C ↾ wf2)
  ∨ x = D ↾ wf4
  → ((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
      ∨ x = D ↾ wf4')
     ∨ x
       ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
           (frames_on_the_way_to_hole'_obligation_5 ψ1
              ψ2 wfψ)
           (frames_on_the_way_to_hole'_obligation_6 p
              wfp)
           (frames_on_the_way_to_hole'_obligation_7 q
              wfq))
    ∨ x
      ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
          (frames_on_the_way_to_hole'_obligation_9 ψ1
             ψ2 wfψ)
          (frames_on_the_way_to_hole'_obligation_10 p
             wfp)
          (frames_on_the_way_to_hole'_obligation_11 q
             wfq)
        intros.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: ((x = A ↾ wf1 ∨ x = B ↾ wf3) ∨ x = C ↾ wf2)
∨ x = D ↾ wf4
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
        destruct_or!.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = A ↾ wf1
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = B ↾ wf3
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = C ↾ wf2
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = D ↾ wf4
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
        {
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = A ↾ wf1
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq) left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = A ↾ wf1
(((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
 ∨ x = D ↾ wf4')
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
      (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_6 p wfp)
      (frames_on_the_way_to_hole'_obligation_7 q wfq) left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = A ↾ wf1
((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
∨ x = D ↾ wf4' left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = A ↾ wf1
(x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3' left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = A ↾ wf1
x = A ↾ wf1' ∨ x = C ↾ wf2' left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = A ↾ wf1
x = A ↾ wf1' pi_assumption. }
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = B ↾ wf3
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = C ↾ wf2
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = D ↾ wf4
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
        {
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = B ↾ wf3
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq) left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = B ↾ wf3
(((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
 ∨ x = D ↾ wf4')
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
      (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_6 p wfp)
      (frames_on_the_way_to_hole'_obligation_7 q wfq) left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = B ↾ wf3
((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
∨ x = D ↾ wf4' left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = B ↾ wf3
(x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3' right.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = B ↾ wf3
x = B ↾ wf3' pi_assumption. }
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = C ↾ wf2
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = D ↾ wf4
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
        {
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = C ↾ wf2
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq) left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = C ↾ wf2
(((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
 ∨ x = D ↾ wf4')
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
      (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_6 p wfp)
      (frames_on_the_way_to_hole'_obligation_7 q wfq) left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = C ↾ wf2
((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
∨ x = D ↾ wf4' left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = C ↾ wf2
(x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3' left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = C ↾ wf2
x = A ↾ wf1' ∨ x = C ↾ wf2' right.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = C ↾ wf2
x = C ↾ wf2' pi_assumption. }
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = D ↾ wf4
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq)
        {
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = D ↾ wf4
((((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
  ∨ x = D ↾ wf4')
 ∨ x
   ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
       (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
          wfψ)
       (frames_on_the_way_to_hole'_obligation_6 p wfp)
       (frames_on_the_way_to_hole'_obligation_7 q wfq))
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ2 p q
      (frames_on_the_way_to_hole'_obligation_9 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_10 p wfp)
      (frames_on_the_way_to_hole'_obligation_11 q wfq) left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = D ↾ wf4
(((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
 ∨ x = D ↾ wf4')
∨ x
  ∈ frames_on_the_way_to_hole' EvS SvS E ψ1 p q
      (frames_on_the_way_to_hole'_obligation_5 ψ1 ψ2
         wfψ)
      (frames_on_the_way_to_hole'_obligation_6 p wfp)
      (frames_on_the_way_to_hole'_obligation_7 q wfq) left.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = D ↾ wf4
((x = A ↾ wf1' ∨ x = C ↾ wf2') ∨ x = B ↾ wf3')
∨ x = D ↾ wf4' right.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 $ ψ2)
EvS: EVarSet
SvS: SVarSet
A: Pattern
wf1: well_formed A = true
C: Pattern
wf2: well_formed C = true
B: Pattern
wf3: well_formed B = true
D: Pattern
wf4: well_formed D = true
wf1': well_formed A = true
wf2': well_formed C = true
wf3': well_formed B = true
wf4': well_formed D = true
x: wfPattern
H: x = D ↾ wf4
x = D ↾ wf4' pi_assumption. }
      }
    }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
wfψ: well_formed ⊥
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ⊥)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ⊥)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ⊥)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ⊥ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ⊥ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ⊥ ⊆ SvS
HEinψ: E ∈ free_evars ⊥
Γ ⊢i ⊥ <---> ⊥ using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
wfψ: well_formed ⊥
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ⊥)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ⊥)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ⊥)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ⊥ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ⊥ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ⊥ ⊆ SvS
HEinψ: E ∈ free_evars ⊥
Γ ⊢i ⊥ <---> ⊥ using gpi
      useBasicReasoning.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
wfψ: well_formed ⊥
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ⊥)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ⊥)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ⊥)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ⊥ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ⊥ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ⊥ ⊆ SvS
HEinψ: E ∈ free_evars ⊥
Γ ⊢i ⊥ <---> ⊥ using BasicReasoning
      apply pf_iff_equiv_refl.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
wfψ: well_formed ⊥
Hsz: 1 ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E ⊥)),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E ⊥)),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E ⊥)
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E ⊥ p
           q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars ⊥ ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars ⊥ ⊆ SvS
HEinψ: E ∈ free_evars ⊥
well_formed ⊥
      abstract (wf_auto2).
    }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      assert (wfψ1 : well_formed ψ1 = true).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
well_formed ψ1 = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
well_formed ψ1 = true clear -wfψ.
Σ: Signature
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
well_formed ψ1 = true abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      assert (size' ψ1 <= sz) by abstract(lia).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      assert (wfψ2 : well_formed ψ2 = true).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
well_formed ψ2 = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
well_formed ψ2 = true clear -wfψ.
Σ: Signature
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
well_formed ψ2 = true abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      assert (size' ψ2 <= sz) by abstract(lia).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi

      pose proof (pf₁ := (IHsz ψ1) ltac:(assumption) ltac:(assumption)).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ∀ (EvS : EVarSet) (SvS : SVarSet),
  ProofInfoLe
    (ExGen := list_to_set
                (evar_fresh_seq EvS
                   (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ1)),
     SVSubst := list_to_set
                  (svar_fresh_seq SvS
                     (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
     KT := if
            decide
              (0 =
               maximal_mu_depth_of_evar_in_pattern'
                 mudepth E ψ1)
           then false
           else true,
     FP := gset_to_coGset
             (frames_on_the_way_to_hole' EvS SvS
                E ψ1 p q wfψ1 wfp wfq)) gpi
  → free_evars p ⊆ EvS
    → free_evars q ⊆ EvS
      → E ∈ EvS
        → free_evars ψ1 ⊆ EvS
          → free_svars p ⊆ SvS
            → free_svars q ⊆ SvS
              → free_svars ψ1 ⊆ SvS
                → Γ
                  ⊢i ψ1^[[evar:E↦p]] <--->
                     ψ1^[[evar:E↦q]] using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      specialize (pf₁ EvS SvS).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      feed specialize pf₁.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E ψ1 p
              q wfψ1 wfp wfq)) gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E ψ1 p
              q wfψ1 wfp wfq)) gpi
        abstract (
          subst i';
          eapply pile_trans;
          [|apply pile];
          apply pile_evs_svs_kt;
          [(simpl; pose proof evar_fresh_seq_max; set_solver)
          |(simpl; pose proof svar_fresh_seq_max; set_solver)
          |(clear pf₁;repeat case_match; simpl; try reflexivity; simpl in *; lia)
          |(apply frames_on_the_way_to_hole'_imp_1)
          ]
        ).

      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS exact p_sub_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS exact q_sub_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
E ∈ EvS exact E_in_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_evars ψ1 ⊆ EvS
        abstract (
          simpl in ψ_sub_EvS;
          clear -ψ_sub_EvS;
          set_solver
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS exact p_sub_SvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS exact q_sub_SvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ1)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ1)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ1)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ1 p q wfψ1 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ1 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ1 ⊆ SvS
              → Γ
                ⊢i ψ1^[[evar:E↦p]] <--->
                   ψ1^[[evar:E↦q]] using gpi
free_svars ψ1 ⊆ SvS abstract (
          simpl in ψ_sub_SvS;
          clear -ψ_sub_SvS;
          set_solver
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi

      pose proof (pf₂ := (IHsz ψ2) ltac:(assumption) ltac:(assumption)).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ∀ (EvS : EVarSet) (SvS : SVarSet),
  ProofInfoLe
    (ExGen := list_to_set
                (evar_fresh_seq EvS
                   (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ2)),
     SVSubst := list_to_set
                  (svar_fresh_seq SvS
                     (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
     KT := if
            decide
              (0 =
               maximal_mu_depth_of_evar_in_pattern'
                 mudepth E ψ2)
           then false
           else true,
     FP := gset_to_coGset
             (frames_on_the_way_to_hole' EvS SvS
                E ψ2 p q wfψ2 wfp wfq)) gpi
  → free_evars p ⊆ EvS
    → free_evars q ⊆ EvS
      → E ∈ EvS
        → free_evars ψ2 ⊆ EvS
          → free_svars p ⊆ SvS
            → free_svars q ⊆ SvS
              → free_svars ψ2 ⊆ SvS
                → Γ
                  ⊢i ψ2^[[evar:E↦p]] <--->
                     ψ2^[[evar:E↦q]] using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi

      specialize (pf₂ EvS SvS).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      feed specialize pf₂.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E ψ2 p
              q wfψ2 wfp wfq)) gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E ψ2 p
              q wfψ2 wfp wfq)) gpi
        abstract (
          subst i';
          eapply pile_trans;
          [|apply pile];
          apply pile_evs_svs_kt;
          [(simpl; rewrite Nat.max_comm; clear; pose proof evar_fresh_seq_max; set_solver)
          |(simpl; rewrite Nat.max_comm; clear; pose proof svar_fresh_seq_max; set_solver)
          |(clear pf₂; repeat case_match; simpl in *; try reflexivity; lia)
          |(apply frames_on_the_way_to_hole'_imp_2)
          ]
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars p ⊆ EvS exact p_sub_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars q ⊆ EvS exact q_sub_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
E ∈ EvS exact E_in_EvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_evars ψ2 ⊆ EvS
        abstract (
          simpl in ψ_sub_EvS;
          clear -ψ_sub_EvS;
          set_solver
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars p ⊆ SvS exact p_sub_SvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars q ⊆ SvS exact q_sub_SvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ2)),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ2)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ2)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              ψ2 p q wfψ2 wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ2 ⊆ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ2 ⊆ SvS
              → Γ
                ⊢i ψ2^[[evar:E↦p]] <--->
                   ψ2^[[evar:E↦q]] using gpi
free_svars ψ2 ⊆ SvS 
        abstract (
          simpl in ψ_sub_SvS;
          clear -ψ_sub_SvS;
          set_solver
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ
⊢i (ψ1^[[evar:E↦p]] ---> ψ2^[[evar:E↦p]]) <--->
   (ψ1^[[evar:E↦q]] ---> ψ2^[[evar:E↦q]]) using gpi

      apply prf_equiv_of_impl_of_equiv.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ1^[[evar:E↦p]] = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ2^[[evar:E↦p]] = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ1^[[evar:E↦q]] = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ2^[[evar:E↦q]] = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ1^[[evar:E↦p]] = true abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ2^[[evar:E↦p]] = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ1^[[evar:E↦q]] = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ2^[[evar:E↦q]] = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ2^[[evar:E↦p]] = true abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ1^[[evar:E↦q]] = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ2^[[evar:E↦q]] = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ1^[[evar:E↦q]] = true abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ2^[[evar:E↦q]] = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
well_formed ψ2^[[evar:E↦q]] = true abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]] using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]] using gpi apply pf₁. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]] using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ1, ψ2: Pattern
wfψ: well_formed (ψ1 ---> ψ2)
Hsz: S (size' ψ1 + size' ψ2) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ψ1 ---> ψ2))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ψ1 ---> ψ2))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ψ1 ---> ψ2))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ψ1 ---> ψ2) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ψ1 ---> ψ2) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ψ1 ---> ψ2) ⊆ SvS
HEinψ: E ∈ free_evars (ψ1 ---> ψ2)
wfψ1: well_formed ψ1 = true
H: size' ψ1 ≤ sz
wfψ2: well_formed ψ2 = true
H0: size' ψ2 ≤ sz
pf₁: Γ ⊢i ψ1^[[evar:E↦p]] <---> ψ1^[[evar:E↦q]]
using gpi
pf₂: Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]]
using gpi
Γ ⊢i ψ2^[[evar:E↦p]] <---> ψ2^[[evar:E↦q]] using gpi apply pf₂. }
    }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      pose proof (frx := set_evar_fresh_is_fresh' EvS).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
frx: evar_fresh (elements EvS) ∉ EvS
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      remember (evar_fresh (elements EvS)) as x.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi

      (* there used to be a destruct on whether E is in psi *)

      assert (well_formed (ψ^{evar: 0 ↦ x})) by abstract(wf_auto2).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      assert (size' (ψ^{evar: 0 ↦ x}) <= sz) by abstract(rewrite evar_open_size'; lia).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi

      pose proof (IH := IHsz (ψ^{evar: 0 ↦ x}) ltac:(assumption) ltac:(assumption)).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ∀ (EvS : EVarSet) (SvS : SVarSet),
  ProofInfoLe
    (ExGen := list_to_set
                (evar_fresh_seq EvS
                   (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ^{evar:0↦x})),
     SVSubst := list_to_set
                  (svar_fresh_seq SvS
                     (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
     KT := if
            decide
              (0 =
               maximal_mu_depth_of_evar_in_pattern'
                 mudepth E ψ^{evar:0↦x})
           then false
           else true,
     FP := gset_to_coGset
             (frames_on_the_way_to_hole' EvS SvS
                E ψ^{evar:0↦x} p q H wfp wfq))
    gpi
  → free_evars p ⊆ EvS
    → free_evars q ⊆ EvS
      → E ∈ EvS
        → free_evars ψ^{evar:0↦x} ⊆ EvS
          → free_svars p ⊆ SvS
            → free_svars q ⊆ SvS
              → free_svars ψ^{evar:0↦x} ⊆ SvS
                → Γ
                  ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                     ψ^{evar:0↦x}^[[evar:E↦q]]
                  using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      specialize (IH ({[x]} ∪ EvS) SvS).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      feed specialize IH.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' ({[x]} ∪ EvS)
              SvS E ψ^{evar:0↦x} p q H wfp wfq)) gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars p ⊆ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars q ⊆ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
E ∈ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars ψ^{evar:0↦x} ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' ({[x]} ∪ EvS)
              SvS E ψ^{evar:0↦x} p q H wfp wfq)) gpi
        abstract(
          subst i';
          simpl;
          eapply pile_trans;
          [|apply pile];
          assert (HxneE: x <> E);
          [(clear -frx E_in_EvS; set_solver)|];
          apply pile_evs_svs_kt;
          [(
            simpl;
            rewrite medoeip_evar_open;
            [(apply not_eq_sym; exact HxneE)
            |(simpl;
              rewrite medoeip_S_in;
              [(assumption)
              |(remember (maximal_exists_depth_of_evar_in_pattern' exdepth E ψ) as n;
                simpl;
                unfold evar_fresh_s;
                rewrite -Heqx;
                clear;
                set_solver)
              ]
            )]
          )
         |(
            simpl;
            rewrite mmdoeip_evar_open;
            [(apply not_eq_sym; exact HxneE)
            |(apply reflexivity)
            ]
          )
        |(
            clear IH;
            repeat case_match; simpl in *; try reflexivity;
            pose proof (Htmp := n);
            rewrite mmdoeip_evar_open in Htmp;
            [(apply not_eq_sym; exact HxneE)|(lia)]
          )
          |(simp frames_on_the_way_to_hole'; subst x; clear; pi_set_solver)
        ]).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars p ⊆ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars q ⊆ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
E ∈ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars ψ^{evar:0↦x} ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars p ⊆ {[x]} ∪ EvS clear -p_sub_EvS.
Σ: Signature
p: Pattern
EvS: EVarSet
p_sub_EvS: free_evars p ⊆ EvS
x: evar
free_evars p ⊆ {[x]} ∪ EvS abstract (set_solver). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars q ⊆ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
E ∈ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars ψ^{evar:0↦x} ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars q ⊆ {[x]} ∪ EvS clear -q_sub_EvS.
Σ: Signature
q: Pattern
EvS: EVarSet
q_sub_EvS: free_evars q ⊆ EvS
x: evar
free_evars q ⊆ {[x]} ∪ EvS abstract (set_solver). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
E ∈ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars ψ^{evar:0↦x} ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
E ∈ {[x]} ∪ EvS clear -E_in_EvS.
Σ: Signature
E: evar
EvS: EVarSet
E_in_EvS: E ∈ EvS
x: evar
E ∈ {[x]} ∪ EvS abstract (set_solver). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars ψ^{evar:0↦x} ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS clear -ψ_sub_EvS.
Σ: Signature
ψ: Pattern
EvS: EVarSet
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
x: evar
free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        abstract (
          rewrite elem_of_subseteq;
          intros x0;
          rewrite free_evars_evar_open'';
          intros [[H1 H2]| H2];
          [
            (subst; clear; set_solver)
            |(simpl in ψ_sub_EvS; set_solver)
          ]
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars ψ^{evar:0↦x} ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ SvS exact p_sub_SvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars ψ^{evar:0↦x} ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ SvS exact q_sub_SvS. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars ψ^{evar:0↦x} ⊆ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq 
                 ({[x]} ∪ EvS)
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{evar:0↦x})),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{evar:0↦x})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{evar:0↦x})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[x]} ∪ EvS) SvS E ψ^{evar:0↦x} p
              q H wfp wfq)) gpi
→ free_evars p ⊆ {[x]} ∪ EvS
  → free_evars q ⊆ {[x]} ∪ EvS
    → E ∈ {[x]} ∪ EvS
      → free_evars ψ^{evar:0↦x} ⊆ {[x]} ∪ EvS
        → free_svars p ⊆ SvS
          → free_svars q ⊆ SvS
            → free_svars ψ^{evar:0↦x} ⊆ SvS
              → Γ
                ⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
                   ψ^{evar:0↦x}^[[evar:E↦q]]
                using gpi
free_svars ψ^{evar:0↦x} ⊆ SvS abstract (
          simpl in ψ_sub_SvS;
          clear -ψ_sub_SvS;
          rewrite free_svars_evar_open;
          exact ψ_sub_SvS
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
Γ ⊢i (ex , ψ^[[evar:E↦p]]) <---> (ex , ψ^[[evar:E↦q]])
using gpi
      eapply congruence_ex_helper with (x := x)(EvS := EvS)(SvS := SvS)(exdepth := exdepth)(mudepth := mudepth); try assumption.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
x ≠ E
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
?i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E 
                  (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E 
                    (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E 
             (ex , ψ))
       then false
       else true, FP := ∅)
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
ProofInfoLe ?i' gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
x ≠ E set_solver. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
?i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E (ex , ψ))
       then false
       else true, FP := ∅)
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
ProofInfoLe ?i' gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
?i' =
(ExGen := list_to_set
            (evar_fresh_seq EvS
               (maximal_exists_depth_of_evar_in_pattern'
                  exdepth E (ex , ψ))),
 SVSubst := list_to_set
              (svar_fresh_seq SvS
                 (maximal_mu_depth_of_evar_in_pattern'
                    mudepth E (ex , ψ))),
 KT := if
        decide
          (0 =
           maximal_mu_depth_of_evar_in_pattern'
             mudepth E (ex , ψ))
       then false
       else true, FP := ∅) reflexivity. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E (ex , ψ))),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E (ex , ψ))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E (ex , ψ))
         then false
         else true, FP := ∅) gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E (ex , ψ))),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E (ex , ψ))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E (ex , ψ))
         then false
         else true, FP := ∅) gpi eapply pile_trans;[|apply pile].
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E (ex , ψ))),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E (ex , ψ))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E (ex , ψ))
         then false
         else true, FP := ∅) i'
        unfold i'.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E (ex , ψ))),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E (ex , ψ))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E (ex , ψ))
         then false
         else true, FP := ∅)
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E (ex , ψ))),
   SVSubst := list_to_set
                (svar_fresh_seq SvS
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E (ex , ψ))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E (ex , ψ))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS SvS E
              (ex , ψ) p q wfψ wfp wfq))
        apply pile_evs_svs_kt.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
list_to_set
  (evar_fresh_seq EvS
     (maximal_exists_depth_of_evar_in_pattern' exdepth
        E (ex , ψ)))
⊆ list_to_set
    (evar_fresh_seq EvS
       (maximal_exists_depth_of_evar_in_pattern'
          exdepth E (ex , ψ)))
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
list_to_set
  (svar_fresh_seq SvS
     (maximal_mu_depth_of_evar_in_pattern' mudepth E
        (ex , ψ)))
⊆ list_to_set
    (svar_fresh_seq SvS
       (maximal_mu_depth_of_evar_in_pattern' mudepth E
          (ex , ψ)))
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ex , ψ))
 then false
 else true) ==>
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ex , ψ))
 then false
 else true)
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
∅
⊆ gset_to_coGset
    (frames_on_the_way_to_hole' EvS SvS E 
       (ex , ψ) p q wfψ wfp wfq)
        {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
list_to_set
  (evar_fresh_seq EvS
     (maximal_exists_depth_of_evar_in_pattern' exdepth
        E (ex , ψ)))
⊆ list_to_set
    (evar_fresh_seq EvS
       (maximal_exists_depth_of_evar_in_pattern'
          exdepth E (ex , ψ))) apply reflexivity. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
list_to_set
  (svar_fresh_seq SvS
     (maximal_mu_depth_of_evar_in_pattern' mudepth E
        (ex , ψ)))
⊆ list_to_set
    (svar_fresh_seq SvS
       (maximal_mu_depth_of_evar_in_pattern' mudepth E
          (ex , ψ)))
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ex , ψ))
 then false
 else true) ==>
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ex , ψ))
 then false
 else true)
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
∅
⊆ gset_to_coGset
    (frames_on_the_way_to_hole' EvS SvS E 
       (ex , ψ) p q wfψ wfp wfq)
        {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
list_to_set
  (svar_fresh_seq SvS
     (maximal_mu_depth_of_evar_in_pattern' mudepth E
        (ex , ψ)))
⊆ list_to_set
    (svar_fresh_seq SvS
       (maximal_mu_depth_of_evar_in_pattern' mudepth E
          (ex , ψ))) apply reflexivity. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ex , ψ))
 then false
 else true) ==>
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ex , ψ))
 then false
 else true)
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
∅
⊆ gset_to_coGset
    (frames_on_the_way_to_hole' EvS SvS E 
       (ex , ψ) p q wfψ wfp wfq)
        {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ex , ψ))
 then false
 else true) ==>
(if
  decide
    (0 =
     maximal_mu_depth_of_evar_in_pattern' mudepth E
       (ex , ψ))
 then false
 else true) case_match; reflexivity. }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
∅
⊆ gset_to_coGset
    (frames_on_the_way_to_hole' EvS SvS E (ex , ψ) p q
       wfψ wfp wfq)
        {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (ex , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (ex , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (ex , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (ex , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (ex , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (ex , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (ex , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (ex , ψ)
x: evar
Heqx: x = evar_fresh (elements EvS)
frx: x ∉ EvS
H: well_formed ψ^{evar:0↦x}
H0: size' ψ^{evar:0↦x} ≤ sz
IH: Γ
⊢i ψ^{evar:0↦x}^[[evar:E↦p]] <--->
   ψ^{evar:0↦x}^[[evar:E↦q]] using gpi
∅
⊆ gset_to_coGset
    (frames_on_the_way_to_hole' EvS SvS E (ex , ψ) p q
       wfψ wfp wfq) clear.
Σ: Signature
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
ψ: Pattern
wfψ: well_formed (ex , ψ)
EvS: EVarSet
SvS: SVarSet
∅
⊆ gset_to_coGset
    (frames_on_the_way_to_hole' EvS SvS E (ex , ψ) p q
       wfψ wfp wfq) set_solver. }
      }
    }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
    {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      pose proof (frX := set_svar_fresh_is_fresh' SvS).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
frX: svar_fresh (elements SvS) ∉ SvS
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      remember (svar_fresh (elements SvS)) as X.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars (mu , ψ)
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi

      simpl in HEinψ.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi

      assert (well_formed (ψ^{svar: 0 ↦ X}) = true) by (abstract(clear -wfψ;wf_auto2)).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      assert (size' (ψ^{svar: 0 ↦ X}) <= sz) by abstract(rewrite svar_open_size'; lia).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      pose proof (IH := IHsz (ψ^{svar: 0 ↦ X}) ltac:(assumption) ltac:(assumption)).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ∀ (EvS : EVarSet) (SvS : SVarSet),
  ProofInfoLe
    (ExGen := list_to_set
                (evar_fresh_seq EvS
                   (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ^{svar:0↦X})),
     SVSubst := list_to_set
                  (svar_fresh_seq SvS
                     (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
     KT := if
            decide
              (0 =
               maximal_mu_depth_of_evar_in_pattern'
                 mudepth E ψ^{svar:0↦X})
           then false
           else true,
     FP := gset_to_coGset
             (frames_on_the_way_to_hole' EvS SvS
                E ψ^{svar:0↦X} p q H wfp wfq))
    gpi
  → free_evars p ⊆ EvS
    → free_evars q ⊆ EvS
      → E ∈ EvS
        → free_evars ψ^{svar:0↦X} ⊆ EvS
          → free_svars p ⊆ SvS
            → free_svars q ⊆ SvS
              → free_svars ψ^{svar:0↦X} ⊆ SvS
                → Γ
                  ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                     ψ^{svar:0↦X}^[[evar:E↦q]]
                  using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      specialize (IH EvS ({[X]} ∪ SvS)).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      feed specialize IH.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H wfp
              wfq)) gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars p ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars ψ^{svar:0↦X} ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars ψ^{svar:0↦X} ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H wfp
              wfq)) gpi
        abstract (
          subst i';
          eapply pile_trans;
          [|apply pile];
          apply pile_evs_svs_kt;
          [
            (
            simpl;
            rewrite medoeip_svar_open;
            apply reflexivity
            )
          |(
            simpl;
            rewrite mmdoeip_svar_open;
            rewrite mmdoeip_S_in;[exact HEinψ|];
            simpl;
            unfold svar_fresh_s;
            rewrite -HeqX;
            clear;
            set_solver
          )
          |(
            clear IH;
            repeat case_match; simpl in *; try reflexivity;
            pose proof (Htmp := n);
            rewrite mmdoeip_svar_open in Htmp;
            pose proof (Htmp2 := e);
            rewrite mmdoeip_S_in in Htmp2;
            [exact HEinψ|];
            inversion Htmp2
          )
          |(simp frames_on_the_way_to_hole'; subst X; clear; pi_set_solver)]).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars p ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars ψ^{svar:0↦X} ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars ψ^{svar:0↦X} ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars p ⊆ EvS
        exact p_sub_EvS.
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars q ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars ψ^{svar:0↦X} ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars ψ^{svar:0↦X} ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars q ⊆ EvS
        exact q_sub_EvS.
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
E ∈ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars ψ^{svar:0↦X} ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars ψ^{svar:0↦X} ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
E ∈ EvS
        exact E_in_EvS.
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars ψ^{svar:0↦X} ⊆ EvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars ψ^{svar:0↦X} ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_evars ψ^{svar:0↦X} ⊆ EvS
        abstract (
          rewrite free_evars_svar_open;
          simpl in ψ_sub_EvS;
          apply ψ_sub_EvS
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars ψ^{svar:0↦X} ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars p ⊆ {[X]} ∪ SvS
        clear -p_sub_SvS.
Σ: Signature
p: Pattern
SvS: SVarSet
p_sub_SvS: free_svars p ⊆ SvS
X: svar
free_svars p ⊆ {[X]} ∪ SvS
        abstract (set_solver).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars ψ^{svar:0↦X} ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars q ⊆ {[X]} ∪ SvS
        clear -q_sub_SvS.
Σ: Signature
q: Pattern
SvS: SVarSet
q_sub_SvS: free_svars q ⊆ SvS
X: svar
free_svars q ⊆ {[X]} ∪ SvS
        abstract (set_solver).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars ψ^{svar:0↦X} ⊆ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
free_svars ψ^{svar:0↦X} ⊆ {[X]} ∪ SvS
        rewrite elem_of_subseteq.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
∀ x : svar,
  x ∈ free_svars ψ^{svar:0↦X} → x ∈ {[X]} ∪ SvS
        intros X'.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
X': svar
X' ∈ free_svars ψ^{svar:0↦X} → X' ∈ {[X]} ∪ SvS
        rewrite free_svars_svar_open''.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
X': svar
X' = X ∧ bsvar_occur ψ 0 ∨ X' ∈ free_svars ψ
→ X' ∈ {[X]} ∪ SvS
        intros [[H1 H2]|H1].
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
X': svar
H1: X' = X
H2: bsvar_occur ψ 0
X' ∈ {[X]} ∪ SvS
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
X': svar
H1: X' ∈ free_svars ψ
X' ∈ {[X]} ∪ SvS
        {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
X': svar
H1: X' = X
H2: bsvar_occur ψ 0
X' ∈ {[X]} ∪ SvS
          abstract (subst X'; clear; set_solver).
        }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
X': svar
H1: X' ∈ free_svars ψ
X' ∈ {[X]} ∪ SvS
        {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq EvS
                 (maximal_exists_depth_of_evar_in_pattern'
                    exdepth E ψ^{svar:0↦X})),
   SVSubst := list_to_set
                (svar_fresh_seq 
                   ({[X]} ∪ SvS)
                   (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ^{svar:0↦X})),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern'
               mudepth E ψ^{svar:0↦X})
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole' EvS
              ({[X]} ∪ SvS) E ψ^{svar:0↦X} p q H
              wfp wfq)) gpi
→ free_evars p ⊆ EvS
  → free_evars q ⊆ EvS
    → E ∈ EvS
      → free_evars ψ^{svar:0↦X} ⊆ EvS
        → free_svars p ⊆ {[X]} ∪ SvS
          → free_svars q ⊆ {[X]} ∪ SvS
            → free_svars ψ^{svar:0↦X}
              ⊆ {[X]} ∪ SvS
              → Γ
                ⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
                   ψ^{svar:0↦X}^[[evar:E↦q]]
                using gpi
X': svar
H1: X' ∈ free_svars ψ
X' ∈ {[X]} ∪ SvS
          abstract (
            simpl in ψ_sub_SvS;
            clear -H1 ψ_sub_SvS;
            set_solver
          ).
        }
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi

      apply pf_iff_free_evar_subst_svar_open_to_bsvar_subst_free_evar_subst in IH.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
well_formed_closed_mu_aux p 0 = true
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
well_formed_closed_mu_aux q 0 = true
      3: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
well_formed_closed_mu_aux q 0 = true clear -wfq.
Σ: Signature
q: Pattern
wfq: well_formed q
well_formed_closed_mu_aux q 0 = true abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
well_formed_closed_mu_aux p 0 = true
      2: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^{svar:0↦X}^[[evar:E↦p]] <--->
   ψ^{svar:0↦X}^[[evar:E↦q]] using gpi
well_formed_closed_mu_aux p 0 = true clear -wfp.
Σ: Signature
p: Pattern
wfp: well_formed p
well_formed_closed_mu_aux p 0 = true abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi

      unshelve (epose proof (IH1 := pf_iff_proj1 _ _ _ _ _ _ IH)).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed ψ^[[evar:E↦p]]^[svar:0↦patt_free_svar X]
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed ψ^[[evar:E↦q]]^[svar:0↦patt_free_svar X]
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed ψ^[[evar:E↦p]]^[svar:0↦patt_free_svar X] clear -wfψ wfp.
Σ: Signature
p: Pattern
wfp: well_formed p
E: evar
ψ: Pattern
wfψ: well_formed (mu , ψ)
X: svar
well_formed ψ^[[evar:E↦p]]^[svar:0↦patt_free_svar X] abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed ψ^[[evar:E↦q]]^[svar:0↦patt_free_svar X]
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed ψ^[[evar:E↦q]]^[svar:0↦patt_free_svar X] clear -wfψ wfq.
Σ: Signature
q: Pattern
wfq: well_formed q
E: evar
ψ: Pattern
wfψ: well_formed (mu , ψ)
X: svar
well_formed ψ^[[evar:E↦q]]^[svar:0↦patt_free_svar X] abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      unshelve (epose proof (IH2 := pf_iff_proj2 _ _ _ _ _ _ IH)).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed ψ^[[evar:E↦p]]^[svar:0↦patt_free_svar X]
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed ψ^[[evar:E↦q]]^[svar:0↦patt_free_svar X]
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed ψ^[[evar:E↦p]]^[svar:0↦patt_free_svar X] clear -wfψ wfp.
Σ: Signature
p: Pattern
wfp: well_formed p
E: evar
ψ: Pattern
wfψ: well_formed (mu , ψ)
X: svar
well_formed ψ^[[evar:E↦p]]^[svar:0↦patt_free_svar X] abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed ψ^[[evar:E↦q]]^[svar:0↦patt_free_svar X]
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed ψ^[[evar:E↦q]]^[svar:0↦patt_free_svar X] clear -wfψ wfq.
Σ: Signature
q: Pattern
wfq: well_formed q
E: evar
ψ: Pattern
wfψ: well_formed (mu , ψ)
X: svar
well_formed ψ^[[evar:E↦q]]^[svar:0↦patt_free_svar X] abstract (wf_auto2). }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ ⊢i (mu , ψ^[[evar:E↦p]]) <---> (mu , ψ^[[evar:E↦q]])
using gpi

      eapply pf_iff_mu_remove_svar_quantify_svar_open.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed_closed_mu_aux ψ^[[evar:E↦p]] 1
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed_closed_mu_aux ψ^[[evar:E↦q]] 1
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
?X ∉ free_svars ψ^[[evar:E↦p]]
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
?X ∉ free_svars ψ^[[evar:E↦q]]
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i (mu , ψ^[[evar:E↦p]]^{svar:0↦?X}^{{svar:?X↦0}}) <--->
   (mu , ψ^[[evar:E↦q]]^{svar:0↦?X}^{{svar:?X↦0}})
using gpi
      5: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i (mu , ψ^[[evar:E↦p]]^{svar:0↦?X}^{{svar:?X↦0}}) <--->
   (mu , ψ^[[evar:E↦q]]^{svar:0↦?X}^{{svar:?X↦0}})
using gpi
        apply pf_iff_split.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed
  (mu , ψ^[[evar:E↦p]]^{svar:0↦?X}^{{svar:?X↦0}})
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed
  (mu , ψ^[[evar:E↦q]]^{svar:0↦?X}^{{svar:?X↦0}})
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i (mu , ψ^[[evar:E↦p]]^{svar:0↦?X}^{{svar:?X↦0}}) --->
   (mu , ψ^[[evar:E↦q]]^{svar:0↦?X}^{{svar:?X↦0}})
using gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i (mu , ψ^[[evar:E↦q]]^{svar:0↦?X}^{{svar:?X↦0}}) --->
   (mu , ψ^[[evar:E↦p]]^{svar:0↦?X}^{{svar:?X↦0}})
using gpi
        4: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i (mu , ψ^[[evar:E↦q]]^{svar:0↦?X}^{{svar:?X↦0}}) --->
   (mu , ψ^[[evar:E↦p]]^{svar:0↦?X}^{{svar:?X↦0}})
using gpi
          apply mu_monotone.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := {[?X]}, KT := true, FP := ∅)
  gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence 
  ?X ψ^[[evar:E↦q]]^{svar:0↦
  ?X} = false
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence 
  ?X ψ^[[evar:E↦p]]^{svar:0↦
  ?X} = false
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i ψ^[[evar:E↦q]]^{svar:0↦
   ?X} ---> ψ^[[evar:E↦p]]^{svar:0↦
   ?X} using gpi
          4: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i ψ^[[evar:E↦q]]^{svar:0↦?X} --->
   ψ^[[evar:E↦p]]^{svar:0↦?X} using gpi
            unfold svar_open.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦patt_free_svar ?X] --->
   ψ^[[evar:E↦p]]^[svar:0↦patt_free_svar ?X] using gpi
            apply IH2.
          }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := {[X]}, KT := true, FP := ∅)
  gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦q]]^{svar:0↦X} = false
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦p]]^{svar:0↦X} = false
          3: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦p]]^{svar:0↦X} = false
            abstract (
              clear -ψ_sub_SvS p_sub_SvS frX wfψ wfp;
              wf_auto2;
              simpl in *;
              pose proof (Htmp := free_svars_free_evar_subst ψ E p);
              clear -Htmp ψ_sub_SvS p_sub_SvS frX;
              set_solver
            ).
          }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := {[X]}, KT := true, FP := ∅)
  gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦q]]^{svar:0↦X} = false
          2: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦q]]^{svar:0↦X} = false
            abstract (
              clear -ψ_sub_SvS q_sub_SvS frX wfψ wfq;
              wf_auto2;
              simpl in *;
              pose proof (Htmp := free_svars_free_evar_subst ψ E q);
              clear -Htmp ψ_sub_SvS q_sub_SvS frX;
              set_solver
            ).
          }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := {[X]}, KT := true, FP := ∅)
  gpi
          {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := {[X]}, KT := true, FP := ∅)
  gpi
            abstract (
              subst i';
              eapply pile_trans;[|apply pile];
              apply pile_evs_svs_kt;
              [(clear; set_solver)
              |(simpl;
                rewrite mmdoeip_S_in;
                [(exact HEinψ)|];
                simpl;
                unfold svar_fresh_s;
                rewrite -HeqX;
                clear;
                set_solver
            )|(repeat case_match; simpl in *; try reflexivity;
            pose proof (Htmp := e);
            rewrite mmdoeip_S_in in Htmp;
            [exact HEinψ|];
            inversion Htmp)|(clear; set_solver)]
            ).
          }
        }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed
  (mu , ψ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}})
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed
  (mu , ψ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i (mu , ψ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}}) --->
   (mu , ψ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
using gpi
        3: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i (mu , ψ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}}) --->
   (mu , ψ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
using gpi
          apply mu_monotone.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := {[X]}, KT := true, FP := ∅)
  gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦p]]^{svar:0↦X} = false
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦q]]^{svar:0↦X} = false
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i ψ^[[evar:E↦p]]^{svar:0↦X} --->
   ψ^[[evar:E↦q]]^{svar:0↦X} using gpi
          4: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i ψ^[[evar:E↦p]]^{svar:0↦X} --->
   ψ^[[evar:E↦q]]^{svar:0↦X} using gpi
            unfold svar_open.
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦patt_free_svar X] using gpi
            apply IH1.
          }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := {[X]}, KT := true, FP := ∅)
  gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦p]]^{svar:0↦X} = false
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦q]]^{svar:0↦X} = false
          3: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦q]]^{svar:0↦X} = false
            abstract (
              clear -ψ_sub_SvS q_sub_SvS frX wfψ wfq;
              wf_auto2; simpl in *;
              pose proof (Htmp := free_svars_free_evar_subst ψ E q);
              clear -Htmp ψ_sub_SvS q_sub_SvS frX;
              set_solver
            ).
          }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := {[X]}, KT := true, FP := ∅)
  gpi
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦p]]^{svar:0↦X} = false
          2: {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
svar_has_negative_occurrence X
  ψ^[[evar:E↦p]]^{svar:0↦X} = false
            abstract (
              clear -ψ_sub_SvS p_sub_SvS frX wfψ wfp;
              wf_auto2; simpl in *;
              pose proof (Htmp := free_svars_free_evar_subst ψ E p);
              clear -Htmp ψ_sub_SvS p_sub_SvS frX;
              set_solver
            ).
          }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := {[X]}, KT := true, FP := ∅)
  gpi
          {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
ProofInfoLe
  (ExGen := ∅, SVSubst := {[X]}, KT := true, FP := ∅)
  gpi
            abstract (
              subst i';
              eapply pile_trans;[|apply pile];
              apply pile_evs_svs_kt;
              [(clear; set_solver)
              |(simpl;
                rewrite mmdoeip_S_in;
                [(exact HEinψ)|];
                simpl;
                unfold svar_fresh_s;
                rewrite -HeqX;
                clear;
                set_solver
              )|(repeat case_match; simpl in *; try reflexivity;
              pose proof (Htmp := e);
              rewrite mmdoeip_S_in in Htmp;
              [exact HEinψ|];
              inversion Htmp)|(clear; set_solver)]
            ).
          }
        }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed
  (mu , ψ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}})
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed
  (mu , ψ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
        {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed
  (mu , ψ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}})
          cut (X ∉ free_svars ψ^[[evar:E↦p]]).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦p]]
→ well_formed
    (mu , ψ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}})
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦p]]
          {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦p]]
→ well_formed
    (mu , ψ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}})
            clear -wfψ wfp.
Σ: Signature
p: Pattern
wfp: well_formed p
E: evar
ψ: Pattern
wfψ: well_formed (mu , ψ)
X: svar
X ∉ free_svars ψ^[[evar:E↦p]]
→ well_formed
    (mu , ψ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}})
            intros.
Σ: Signature
p: Pattern
wfp: well_formed p
E: evar
ψ: Pattern
wfψ: well_formed (mu , ψ)
X: svar
H: X ∉ free_svars ψ^[[evar:E↦p]]
well_formed
  (mu , ψ^[[evar:E↦p]]^{svar:0↦X}^{{svar:X↦0}})
            (* TODO: this rewrite somewhy does not happen in wf_auto2 *)
            abstract (rewrite svar_quantify_svar_open;wf_auto2).
          }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦p]]
          abstract (
            pose proof (Htmp := free_svars_free_evar_subst ψ E p);
            clear -H Htmp frX ψ_sub_SvS p_sub_SvS;
            set_solver
          ).
        }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed
  (mu , ψ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
        {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed
  (mu , ψ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
          cut (X ∉ free_svars ψ^[[evar:E↦q]]).
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦q]]
→ well_formed
    (mu , ψ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦q]]
          {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦q]]
→ well_formed
    (mu , ψ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
            clear -wfψ wfq.
Σ: Signature
q: Pattern
wfq: well_formed q
E: evar
ψ: Pattern
wfψ: well_formed (mu , ψ)
X: svar
X ∉ free_svars ψ^[[evar:E↦q]]
→ well_formed
    (mu , ψ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
            intros.
Σ: Signature
q: Pattern
wfq: well_formed q
E: evar
ψ: Pattern
wfψ: well_formed (mu , ψ)
X: svar
H: X ∉ free_svars ψ^[[evar:E↦q]]
well_formed
  (mu , ψ^[[evar:E↦q]]^{svar:0↦X}^{{svar:X↦0}})
            abstract (rewrite svar_quantify_svar_open; wf_auto2).
          }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦q]]
          abstract (
            pose proof (Htmp := free_svars_free_evar_subst ψ E q);
            clear -H Htmp frX ψ_sub_SvS q_sub_SvS;
            set_solver
          ).
        }
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed_closed_mu_aux ψ^[[evar:E↦p]] 1
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed_closed_mu_aux ψ^[[evar:E↦q]] 1
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦p]]
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦q]]
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed_closed_mu_aux ψ^[[evar:E↦p]] 1
        clear -wfψ wfp.
Σ: Signature
p: Pattern
wfp: well_formed p
E: evar
ψ: Pattern
wfψ: well_formed (mu , ψ)
well_formed_closed_mu_aux ψ^[[evar:E↦p]] 1
        abstract (wf_auto2).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed_closed_mu_aux ψ^[[evar:E↦q]] 1
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦p]]
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦q]]
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
well_formed_closed_mu_aux ψ^[[evar:E↦q]] 1
        clear -wfψ wfq.
Σ: Signature
q: Pattern
wfq: well_formed q
E: evar
ψ: Pattern
wfψ: well_formed (mu , ψ)
well_formed_closed_mu_aux ψ^[[evar:E↦q]] 1
        abstract (wf_auto2).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦p]]
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦q]]
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦p]]
        abstract (
          pose proof (Htmp := free_svars_free_evar_subst ψ E p);
          clear -H Htmp frX ψ_sub_SvS p_sub_SvS;
          set_solver
        ).
      }
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦q]]
      {
Σ: Signature
sz: nat
Γ: Theory
p, q: Pattern
wfp: well_formed p
wfq: well_formed q
E: evar
exdepth, mudepth: nat
gpi: ProofInfo
pf: Γ ⊢i p <---> q using gpi
IHsz: ∀ (ψ : Pattern) (wfψ : well_formed ψ),
  size' ψ ≤ sz
  → ∀ (EvS : EVarSet) (SvS : SVarSet),
      ProofInfoLe
        (ExGen := list_to_set
                    (evar_fresh_seq EvS
                      (maximal_exists_depth_of_evar_in_pattern'
                      exdepth E ψ)),
         SVSubst := list_to_set
                      (svar_fresh_seq SvS
                      (maximal_mu_depth_of_evar_in_pattern'
                      mudepth E ψ)),
         KT := if
                decide
                  (0 =
                   maximal_mu_depth_of_evar_in_pattern'
                     mudepth E ψ)
               then false
               else true,
         FP := gset_to_coGset
                 (frames_on_the_way_to_hole'
                    EvS SvS E ψ p q wfψ wfp wfq))
        gpi
      → free_evars p ⊆ EvS
        → free_evars q ⊆ EvS
          → E ∈ EvS
            → free_evars ψ ⊆ EvS
              → free_svars p ⊆ SvS
                → free_svars q ⊆ SvS
                  → free_svars ψ ⊆ SvS
                    → Γ
                      ⊢i 
                      ψ^[[evar:E↦p]] <--->
                      ψ^[[evar:E↦q]] using gpi
ψ: Pattern
wfψ: well_formed (mu , ψ)
Hsz: S (size' ψ) ≤ S sz
EvS: EVarSet
SvS: SVarSet
i':= ExGen := list_to_set
           (evar_fresh_seq EvS
              (maximal_exists_depth_of_evar_in_pattern'
                 exdepth E 
                 (mu , ψ))),
SVSubst := list_to_set
             (svar_fresh_seq SvS
                (maximal_mu_depth_of_evar_in_pattern'
                   mudepth E 
                   (mu , ψ))),
KT := if
       decide
         (0 =
          maximal_mu_depth_of_evar_in_pattern'
            mudepth E (mu , ψ))
      then false
      else true,
FP := gset_to_coGset
        (frames_on_the_way_to_hole' EvS SvS E
           (mu , ψ) p q wfψ wfp wfq): ProofInfo
pile: ProofInfoLe i' gpi
p_sub_EvS: free_evars p ⊆ EvS
q_sub_EvS: free_evars q ⊆ EvS
E_in_EvS: E ∈ EvS
ψ_sub_EvS: free_evars (mu , ψ) ⊆ EvS
p_sub_SvS: free_svars p ⊆ SvS
q_sub_SvS: free_svars q ⊆ SvS
ψ_sub_SvS: free_svars (mu , ψ) ⊆ SvS
HEinψ: E ∈ free_evars ψ
X: svar
HeqX: X = svar_fresh (elements SvS)
frX: X ∉ SvS
H: well_formed ψ^{svar:0↦X} = true
H0: size' ψ^{svar:0↦X} ≤ sz
IH: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] <--->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH1: Γ
⊢i ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] using gpi
IH2: Γ
⊢i ψ^[[evar:E↦q]]^[svar:0↦
   patt_free_svar X] --->
   ψ^[[evar:E↦p]]^[svar:0↦
   patt_free_svar X] using gpi
X ∉ free_svars ψ^[[evar:E↦q]]
        abstract (
          pose proof (Htmp := free_svars_free_evar_subst ψ E q);
          clear -H Htmp frX ψ_sub_SvS q_sub_SvS;
          set_solver
        ).
      }
    }
  Defined.

  Lemma prf_equiv_congruence Γ p q C
    (gpi : ProofInfo)
    (wfp : well_formed p = true)
    (wfq : well_formed q = true)
    (wfC: PC_wf C)
    (pile : ProofInfoLe
     (
       (ExGen := list_to_set (evar_fresh_seq (free_evars (pcPattern C) ∪ free_evars p ∪ free_evars q ∪ {[pcEvar C]}) (maximal_exists_depth_of_evar_in_pattern (pcEvar C) (pcPattern C))),
       SVSubst := list_to_set (svar_fresh_seq (free_svars (pcPattern C) ∪ free_svars p ∪ free_svars q) (maximal_mu_depth_of_evar_in_pattern (pcEvar C) (pcPattern C))),
       KT := if decide (0 = (maximal_mu_depth_of_evar_in_pattern (pcEvar C) (pcPattern C))) is left _ then false else true,
       FP := gset_to_coGset (@frames_on_the_way_to_hole' Σ (free_evars (pcPattern C) ∪ free_evars p ∪ free_evars q ∪ {[pcEvar C]}) (free_svars (pcPattern C) ∪ free_svars p ∪ free_svars q) (pcEvar C) (pcPattern C) p q wfC wfp wfq))
      )
     ( gpi)
    ):
      Γ ⊢i (p <---> q) using ( gpi) ->
      Γ ⊢i (((emplace C p) <---> (emplace C q))) using ( gpi).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf C
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
Γ ⊢i p <---> q using gpi
→ Γ ⊢i (C [p]) <---> (C [q]) using gpi
  Proof.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf C
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
Γ ⊢i p <---> q using gpi
→ Γ ⊢i (C [p]) <---> (C [q]) using gpi
    intros Hiff.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf C
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
Hiff: Γ ⊢i p <---> q using gpi
Γ ⊢i (C [p]) <---> (C [q]) using gpi
    assert (well_formed (p <---> q)).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf C
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
Hiff: Γ ⊢i p <---> q using gpi
well_formed (p <---> q)
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf C
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
Γ ⊢i (C [p]) <---> (C [q]) using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf C
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
Hiff: Γ ⊢i p <---> q using gpi
well_formed (p <---> q) abstract (
        pose proof (proved_impl_wf _ _ (proj1_sig Hiff));
        assumption
      ).
    }
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf C
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
Γ ⊢i (C [p]) <---> (C [q]) using gpi
    assert (well_formed p) by (abstract (wf_auto2)).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf C
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
Γ ⊢i (C [p]) <---> (C [q]) using gpi
    assert (well_formed q) by (abstract (wf_auto2)).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf C
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i (C [p]) <---> (C [q]) using gpi
    destruct C as [E ψ].
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars
                    (pcPattern
                       {|
                        pcEvar := E;
                        pcPattern := ψ
                       |}) ∪ 
                  free_evars p ∪ 
                  free_evars q
                  ∪ {[pcEvar
                        {|
                        pcEvar := E;
                        pcPattern := ψ
                        |}]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar
                       {|
                        pcEvar := E;
                        pcPattern := ψ
                       |})
                    (pcPattern
                       {|
                        pcEvar := E;
                        pcPattern := ψ
                       |}))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars
                      (pcPattern
                        {|
                        pcEvar := E;
                        pcPattern := ψ
                        |}) ∪ 
                    free_svars p ∪ 
                    free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar
                        {|
                        pcEvar := E;
                        pcPattern := ψ
                        |})
                      (pcPattern
                        {|
                        pcEvar := E;
                        pcPattern := ψ
                        |}))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar
                  {|
                    pcEvar := E; pcPattern := ψ
                  |})
               (pcPattern
                  {|
                    pcEvar := E; pcPattern := ψ
                  |}))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars
                 (pcPattern
                    {|
                      pcEvar := E;
                      pcPattern := ψ
                    |}) ∪ 
               free_evars p ∪ 
               free_evars q
               ∪ {[pcEvar
                     {|
                       pcEvar := E;
                       pcPattern := ψ
                     |}]})
              (free_svars
                 (pcPattern
                    {|
                      pcEvar := E;
                      pcPattern := ψ
                    |}) ∪ 
               free_svars p ∪ 
               free_svars q)
              (pcEvar
                 {|
                   pcEvar := E; pcPattern := ψ
                 |})
              (pcPattern
                 {|
                   pcEvar := E; pcPattern := ψ
                 |}) p q wfC wfp wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ
⊢i ({| pcEvar := E; pcPattern := ψ |} [p]) <--->
   ({| pcEvar := E; pcPattern := ψ |} [q]) using gpi simpl in *.
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ
⊢i ({| pcEvar := E; pcPattern := ψ |} [p]) <--->
   ({| pcEvar := E; pcPattern := ψ |} [q]) using gpi
    unfold emplace.
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ
⊢i (pcPattern {| pcEvar := E; pcPattern := ψ |})^[[evar:
   pcEvar {| pcEvar := E; pcPattern := ψ |}↦p]] <--->
   (pcPattern {| pcEvar := E; pcPattern := ψ |})^[[evar:
   pcEvar {| pcEvar := E; pcPattern := ψ |}↦q]]
using gpi simpl.
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i ψ^[[evar:E↦p]] <---> ψ^[[evar:E↦q]] using gpi
    eapply eq_prf_equiv_congruence
    with (EvS := (free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}))
         (SvS := (free_svars ψ ∪ free_svars p ∪ free_svars q))
         (exdepth := 0)
         (mudepth := 0)
         (sz := size' ψ)
    .
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
size' ψ ≤ size' ψ
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars p
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars q
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars ψ
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
E ∈ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars p
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars q
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars ψ
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern'
                    0 E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern'
                      0 E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern' 0 E
               ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q 
              ?wfψ ?wfp ?wfq)) gpi
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i p <---> q using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
size' ψ ≤ size' ψ apply reflexivity. }
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars p
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars q
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars ψ
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
E ∈ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars p
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars q
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars ψ
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern'
                    0 E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern'
                      0 E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern' 0 E
               ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q 
              ?wfψ ?wfp ?wfq)) gpi
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i p <---> q using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars p
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]} abstract (clear; set_solver). }
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars q
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars ψ
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
E ∈ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars p
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars q
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars ψ
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern'
                    0 E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern'
                      0 E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern' 0 E
               ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q 
              ?wfψ ?wfp ?wfq)) gpi
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i p <---> q using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars q
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]} abstract (clear; set_solver). }
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars ψ
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
E ∈ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars p
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars q
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars ψ
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern'
                    0 E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern'
                      0 E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern' 0 E
               ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q 
              ?wfψ ?wfp ?wfq)) gpi
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i p <---> q using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_evars ψ
⊆ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]} abstract (clear; set_solver). }
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
E ∈ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]}
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars p
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars q
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars ψ
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern'
                    0 E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern'
                      0 E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern' 0 E
               ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q 
              ?wfψ ?wfp ?wfq)) gpi
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i p <---> q using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
E ∈ free_evars ψ ∪ free_evars p ∪ free_evars q ∪ {[E]} abstract (clear; set_solver). }
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars p
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars q
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars ψ
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern'
                    0 E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern'
                      0 E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern' 0 E
               ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q 
              ?wfψ ?wfp ?wfq)) gpi
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i p <---> q using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars p
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q abstract (clear; set_solver). }
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars q
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars ψ
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern'
                    0 E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern'
                      0 E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern' 0 E
               ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q 
              ?wfψ ?wfp ?wfq)) gpi
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i p <---> q using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars q
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q abstract (clear; set_solver). }
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars ψ
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern'
                    0 E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern'
                      0 E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern' 0 E
               ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q 
              ?wfψ ?wfp ?wfq)) gpi
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i p <---> q using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
free_svars ψ
⊆ free_svars ψ ∪ free_svars p ∪ free_svars q abstract (clear; set_solver). }
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern'
                    0 E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern'
                      0 E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern' 0 E
               ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q ?wfψ ?wfp ?wfq))
  gpi
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i p <---> q using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern'
                    0 E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern'
                      0 E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern' 0 E
               ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q ?wfψ ?wfp ?wfq))
  gpi eapply pile_trans;[|apply pile].
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern'
                    0 E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern'
                      0 E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern' 0 E
               ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q ?wfψ ?wfp ?wfq))
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp wfq)) apply pile_refl. }
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i p <---> q using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
E: evar
ψ: Pattern
gpi: ProofInfo
wfp: well_formed p = true
wfq: well_formed q = true
wfC: PC_wf {| pcEvar := E; pcPattern := ψ |}
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars ψ ∪ free_evars p
                  ∪ free_evars q ∪ {[E]})
                 (maximal_exists_depth_of_evar_in_pattern
                    E ψ)),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars ψ ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      E ψ)),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               E ψ)
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars ψ ∪ free_evars p
               ∪ free_evars q ∪ {[E]})
              (free_svars ψ ∪ free_svars p
               ∪ free_svars q) E ψ p q wfC wfp
              wfq)) gpi
Hiff: Γ ⊢i p <---> q using gpi
H: well_formed (p <---> q)
H0: well_formed p
H1: well_formed q
Γ ⊢i p <---> q using gpi exact Hiff. }
  Defined.

End FOL_helpers.


Lemma prf_equiv_congruence_iter {Σ : Signature} (Γ : Theory) (p q : Pattern) (C : PatternCtx) l
  (wfp : well_formed p)
  (wfq : well_formed q)
  (wfC : PC_wf C)
  (gpi : ProofInfo)
  (pile : ProofInfoLe
  (
    (ExGen := list_to_set (evar_fresh_seq (free_evars (pcPattern C) ∪ free_evars p ∪ free_evars q ∪ {[pcEvar C]}) (maximal_exists_depth_of_evar_in_pattern (pcEvar C) (pcPattern C))),
      SVSubst := list_to_set (svar_fresh_seq (free_svars (pcPattern C) ∪ free_svars p ∪ free_svars q) (maximal_mu_depth_of_evar_in_pattern (pcEvar C) (pcPattern C))),
      KT := if decide (0 = (maximal_mu_depth_of_evar_in_pattern (pcEvar C) (pcPattern C))) is left _ then false else true,
      FP := gset_to_coGset (@frames_on_the_way_to_hole' Σ (free_evars (pcPattern C) ∪ free_evars p ∪ free_evars q ∪ {[pcEvar C]}) (free_svars (pcPattern C) ∪ free_svars p ∪ free_svars q) (pcEvar C) (pcPattern C) p q wfC wfp wfq))  
    )
    ( gpi)
  ):
  Pattern.wf l ->
  Γ ⊢i p <---> q using ( gpi) ->
  Γ ⊢i (foldr patt_imp (emplace C p) l) <---> (foldr patt_imp (emplace C q) l) using ( gpi).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wf l
→ Γ ⊢i p <---> q using gpi
  → Γ
    ⊢i foldr patt_imp (C [p]) l <--->
       foldr patt_imp (C [q]) l using gpi
Proof.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wf l
→ Γ ⊢i p <---> q using gpi
  → Γ
    ⊢i foldr patt_imp (C [p]) l <--->
       foldr patt_imp (C [q]) l using gpi
  intros wfl Himp.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfl: wf l
Himp: Γ ⊢i p <---> q using gpi
Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
  induction l; simpl in *.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfl: wf []
Himp: Γ ⊢i p <---> q using gpi
Γ ⊢i (C [p]) <---> (C [q]) using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfl: wf (a :: l)
Himp: Γ ⊢i p <---> q using gpi
IHl: wf l
→ Γ
  ⊢i foldr patt_imp (C [p]) l <--->
     foldr patt_imp (C [q]) l using gpi
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi
  -
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfl: wf []
Himp: Γ ⊢i p <---> q using gpi
Γ ⊢i (C [p]) <---> (C [q]) using gpi unshelve(eapply prf_equiv_congruence); assumption.
  -
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfl: wf (a :: l)
Himp: Γ ⊢i p <---> q using gpi
IHl: wf l
→ Γ
  ⊢i foldr patt_imp (C [p]) l <--->
     foldr patt_imp (C [q]) l using gpi
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi pose proof (wfal := wfl).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfl: wf (a :: l)
Himp: Γ ⊢i p <---> q using gpi
IHl: wf l
→ Γ
  ⊢i foldr patt_imp (C [p]) l <--->
     foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi
    unfold Pattern.wf in wfl.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfl: foldr andb true (map well_formed (a :: l))
Himp: Γ ⊢i p <---> q using gpi
IHl: wf l
→ Γ
  ⊢i foldr patt_imp (C [p]) l <--->
     foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi simpl in wfl.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfl: well_formed a &&
foldr andb true (map well_formed l)
Himp: Γ ⊢i p <---> q using gpi
IHl: wf l
→ Γ
  ⊢i foldr patt_imp (C [p]) l <--->
     foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi apply andb_prop in wfl as [wfa wfl].
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: wf l
→ Γ
  ⊢i foldr patt_imp (C [p]) l <--->
     foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi
    specialize (IHl wfl).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi
    pose proof (Hwf1 := proved_impl_wf _ _ (proj1_sig IHl)).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi
    pose proof (Hwf2 := proved_impl_wf _ _ (proj1_sig Himp)).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi
    assert (well_formed (emplace C p)).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
well_formed (C [p])
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
well_formed (C [p])
      unfold emplace.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
well_formed (pcPattern C)^[[evar:pcEvar C↦p]]
      wf_auto2.
    }
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi
    assert (well_formed (emplace C q)).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
well_formed (C [q])
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi
    {
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
well_formed (C [q])
      unfold emplace.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
well_formed (pcPattern C)^[[evar:pcEvar C↦q]]
      wf_auto2.
    }
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ
⊢i (a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l) using gpi
    toMLGoal.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
well_formed
  ((a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l))
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ Ⱶ
(a ---> foldr patt_imp (C [p]) l) <--->
(a ---> foldr patt_imp (C [q]) l)
using gpi

    {
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
well_formed
  ((a ---> foldr patt_imp (C [p]) l) <--->
   (a ---> foldr patt_imp (C [q]) l)) unfold emplace.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
well_formed
  ((a --->
    foldr patt_imp (pcPattern C)^[[evar:pcEvar C↦p]] l) <--->
   (a --->
    foldr patt_imp (pcPattern C)^[[evar:pcEvar C↦q]] l)) wf_auto2. }
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ Ⱶ
(a ---> foldr patt_imp (C [p]) l) <--->
(a ---> foldr patt_imp (C [q]) l)
using gpi

    unfold patt_iff.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ Ⱶ
((a ---> foldr patt_imp (C [p]) l) --->
 a ---> foldr patt_imp (C [q]) l)
and ((a ---> foldr patt_imp (C [q]) l) --->
     a ---> foldr patt_imp (C [p]) l)
using gpi

    mlSplitAnd.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ Ⱶ
(a ---> foldr patt_imp (C [p]) l) --->
a ---> foldr patt_imp (C [q]) l
using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ Ⱶ
(a ---> foldr patt_imp (C [q]) l) --->
a ---> foldr patt_imp (C [p]) l
using gpi

    +
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ Ⱶ
(a ---> foldr patt_imp (C [p]) l) --->
a ---> foldr patt_imp (C [q]) l
using gpi
 mlIntro.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
--------------------------------------
a ---> foldr patt_imp (C [q]) l
using gpi
 mlIntro.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
"1" ∶ a,
--------------------------------------
foldr patt_imp (C [q]) l
using gpi

      mlAssert ((foldr patt_imp (emplace C p) l)).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
well_formed (foldr patt_imp (C [p]) l)
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
"1" ∶ a,
--------------------------------------
foldr patt_imp (C [p]) l
using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [p]) l,
--------------------------------------
foldr patt_imp (C [q]) l
using gpi

      {
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
well_formed (foldr patt_imp (C [p]) l) wf_auto2. }
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
"1" ∶ a,
--------------------------------------
foldr patt_imp (C [p]) l
using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [p]) l,
--------------------------------------
foldr patt_imp (C [q]) l
using gpi

      {
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
"1" ∶ a,
--------------------------------------
foldr patt_imp (C [p]) l
using gpi
 mlApply "0".
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
"1" ∶ a,
--------------------------------------
a
using gpi
 mlExactn 1. }
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [p]) l,
--------------------------------------
foldr patt_imp (C [q]) l
using gpi

      apply pf_iff_proj1 in IHl.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l --->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [p]) l,
--------------------------------------
foldr patt_imp (C [q]) l
using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
well_formed (foldr patt_imp (C [p]) l)
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
well_formed (foldr patt_imp (C [q]) l)
      2,3: wf_auto2.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l --->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [p]) l,
--------------------------------------
foldr patt_imp (C [q]) l
using gpi

      mlApplyMetaRaw IHl.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l --->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [p]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [p]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [p]) l,
--------------------------------------
foldr patt_imp (C [p]) l
using gpi

      mlExactn 2.
    +
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ Ⱶ
(a ---> foldr patt_imp (C [q]) l) --->
a ---> foldr patt_imp (C [p]) l
using gpi
 mlIntro.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
--------------------------------------
a ---> foldr patt_imp (C [p]) l
using gpi
 mlIntro.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
"1" ∶ a,
--------------------------------------
foldr patt_imp (C [p]) l
using gpi

      mlAssert ((foldr patt_imp (emplace C q) l)).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
well_formed (foldr patt_imp (C [q]) l)
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
"1" ∶ a,
--------------------------------------
foldr patt_imp (C [q]) l
using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [q]) l,
--------------------------------------
foldr patt_imp (C [p]) l
using gpi

      {
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
well_formed (foldr patt_imp (C [q]) l) wf_auto2. }
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
"1" ∶ a,
--------------------------------------
foldr patt_imp (C [q]) l
using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [q]) l,
--------------------------------------
foldr patt_imp (C [p]) l
using gpi

      {
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
"1" ∶ a,
--------------------------------------
foldr patt_imp (C [q]) l
using gpi
 mlApply "0".
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
"1" ∶ a,
--------------------------------------
a
using gpi
 mlExactn 1. }
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [q]) l,
--------------------------------------
foldr patt_imp (C [p]) l
using gpi

      apply pf_iff_proj2 in IHl.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [q]) l --->
   foldr patt_imp (C [p]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [q]) l,
--------------------------------------
foldr patt_imp (C [p]) l
using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
well_formed (foldr patt_imp (C [p]) l)
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
well_formed (foldr patt_imp (C [q]) l)
      2,3: wf_auto2.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [q]) l --->
   foldr patt_imp (C [p]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [q]) l,
--------------------------------------
foldr patt_imp (C [p]) l
using gpi

      mlApplyMetaRaw IHl.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
a: Pattern
l: list Pattern
wfp: well_formed p
wfq: well_formed q
wfC: PC_wf C
gpi: ProofInfo
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC wfp wfq))
  gpi
wfa: well_formed a = true
wfl: foldr andb true (map well_formed l) = true
Himp: Γ ⊢i p <---> q using gpi
IHl: Γ
⊢i foldr patt_imp (C [q]) l --->
   foldr patt_imp (C [p]) l using gpi
wfal: wf (a :: l)
Hwf1: well_formed
  (foldr patt_imp (C [p]) l <--->
   foldr patt_imp (C [q]) l)
Hwf2: well_formed (p <---> q)
H: well_formed (C [p])
H0: well_formed (C [q])
Hwf: well_formed (foldr patt_imp (C [q]) l)
Γ Ⱶ
"0" ∶ a ---> foldr patt_imp (C [q]) l,
"1" ∶ a,
"2" ∶ foldr patt_imp (C [q]) l,
--------------------------------------
foldr patt_imp (C [q]) l
using gpi

      mlExactn 2.
Defined.

Lemma extract_wfp {Σ : Signature} (Γ : Theory) (p q : Pattern) (i : ProofInfo):
  Γ ⊢i p <---> q using i ->
  well_formed p.
Σ: Signature
Γ: Theory
p, q: Pattern
i: ProofInfo
Γ ⊢i p <---> q using i → well_formed p
Proof.
Σ: Signature
Γ: Theory
p, q: Pattern
i: ProofInfo
Γ ⊢i p <---> q using i → well_formed p
  intros H.
Σ: Signature
Γ: Theory
p, q: Pattern
i: ProofInfo
H: Γ ⊢i p <---> q using i
well_formed p
  pose proof (H' := proj1_sig H).
Σ: Signature
Γ: Theory
p, q: Pattern
i: ProofInfo
H: Γ ⊢i p <---> q using i
H': ML_proof_system Γ (p <---> q)
well_formed p
  apply proved_impl_wf in H'.
Σ: Signature
Γ: Theory
p, q: Pattern
i: ProofInfo
H: Γ ⊢i p <---> q using i
H': well_formed (p <---> q)
well_formed p
  wf_auto2.
Qed.

Lemma extract_wfq {Σ : Signature} (Γ : Theory) (p q : Pattern) (i : ProofInfo):
  Γ ⊢i p <---> q using i ->
  well_formed q.
Σ: Signature
Γ: Theory
p, q: Pattern
i: ProofInfo
Γ ⊢i p <---> q using i → well_formed q
Proof.
Σ: Signature
Γ: Theory
p, q: Pattern
i: ProofInfo
Γ ⊢i p <---> q using i → well_formed q
  intros H.
Σ: Signature
Γ: Theory
p, q: Pattern
i: ProofInfo
H: Γ ⊢i p <---> q using i
well_formed q
  pose proof (H' := proj1_sig H).
Σ: Signature
Γ: Theory
p, q: Pattern
i: ProofInfo
H: Γ ⊢i p <---> q using i
H': ML_proof_system Γ (p <---> q)
well_formed q
  apply proved_impl_wf in H'.
Σ: Signature
Γ: Theory
p, q: Pattern
i: ProofInfo
H: Γ ⊢i p <---> q using i
H': well_formed (p <---> q)
well_formed q
  wf_auto2.
Qed.

Lemma MLGoal_rewriteIff
  {Σ : Signature} (Γ : Theory) (p q : Pattern) (C : PatternCtx) l (gpi : ProofInfo)
  (wfC : PC_wf C)
  (pf : Γ ⊢i p <---> q using ( gpi)) :
  mkMLGoal Σ Γ l (emplace C q) ( gpi) ->
  (ProofInfoLe
  (
     (ExGen := list_to_set
                 (evar_fresh_seq
                    (free_evars (pcPattern C) ∪ free_evars p ∪ free_evars q
                     ∪ {[pcEvar C]})
                    (maximal_exists_depth_of_evar_in_pattern 
                       (pcEvar C) (pcPattern C))),
      SVSubst := list_to_set
                   (svar_fresh_seq
                      (free_svars (pcPattern C) ∪ free_svars p
                       ∪ free_svars q)
                      (maximal_mu_depth_of_evar_in_pattern 
                         (pcEvar C) (pcPattern C))),
      KT := (if
              decide
                (0 =
                 maximal_mu_depth_of_evar_in_pattern (pcEvar C) (pcPattern C))
             is left _
             then false
             else true
             ),
      FP := gset_to_coGset (@frames_on_the_way_to_hole' Σ (free_evars (pcPattern C) ∪ free_evars p ∪ free_evars q ∪ {[pcEvar C]}) (free_svars (pcPattern C) ∪ free_svars p ∪ free_svars q) (pcEvar C) (pcPattern C) p q wfC (extract_wfp Γ p q ( gpi) pf) (extract_wfq Γ p q ( gpi) pf))
      ))
      ( gpi)) ->
  mkMLGoal Σ Γ l (emplace C p) ( gpi).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
pf: Γ ⊢i p <---> q using gpi
Γ Ⱶ
l--------------------------------------
C [q]
using gpi

→ ProofInfoLe
    (ExGen := list_to_set
                (evar_fresh_seq
                   (free_evars (pcPattern C)
                    ∪ free_evars p ∪ free_evars q
                    ∪ {[pcEvar C]})
                   (maximal_exists_depth_of_evar_in_pattern
                      (pcEvar C) (pcPattern C))),
     SVSubst := list_to_set
                  (svar_fresh_seq
                     (free_svars (pcPattern C)
                      ∪ free_svars p ∪ free_svars q)
                     (maximal_mu_depth_of_evar_in_pattern
                        (pcEvar C) (pcPattern C))),
     KT := if
            decide
              (0 =
               maximal_mu_depth_of_evar_in_pattern
                 (pcEvar C) (pcPattern C))
           then false
           else true,
     FP := gset_to_coGset
             (frames_on_the_way_to_hole'
                (free_evars (pcPattern C)
                 ∪ free_evars p ∪ free_evars q
                 ∪ {[pcEvar C]})
                (free_svars (pcPattern C)
                 ∪ free_svars p ∪ free_svars q)
                (pcEvar C) (pcPattern C) p q wfC
                (extract_wfp Γ p q gpi pf)
                (extract_wfq Γ p q gpi pf))) gpi
  → Γ Ⱶ
  l--------------------------------------
  C [p]
  using gpi
  
Proof.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
pf: Γ ⊢i p <---> q using gpi
Γ Ⱶ
l--------------------------------------
C [q]
using gpi

→ ProofInfoLe
    (ExGen := list_to_set
                (evar_fresh_seq
                   (free_evars (pcPattern C)
                    ∪ free_evars p ∪ free_evars q
                    ∪ {[pcEvar C]})
                   (maximal_exists_depth_of_evar_in_pattern
                      (pcEvar C) (pcPattern C))),
     SVSubst := list_to_set
                  (svar_fresh_seq
                     (free_svars (pcPattern C)
                      ∪ free_svars p ∪ free_svars q)
                     (maximal_mu_depth_of_evar_in_pattern
                        (pcEvar C) (pcPattern C))),
     KT := if
            decide
              (0 =
               maximal_mu_depth_of_evar_in_pattern
                 (pcEvar C) (pcPattern C))
           then false
           else true,
     FP := gset_to_coGset
             (frames_on_the_way_to_hole'
                (free_evars (pcPattern C)
                 ∪ free_evars p ∪ free_evars q
                 ∪ {[pcEvar C]})
                (free_svars (pcPattern C)
                 ∪ free_svars p ∪ free_svars q)
                (pcEvar C) (pcPattern C) p q wfC
                (extract_wfp Γ p q gpi pf)
                (extract_wfq Γ p q gpi pf))) gpi
  → Γ Ⱶ
  l--------------------------------------
  C [p]
  using gpi
  
  rename pf into Hpiffq.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
Γ Ⱶ
l--------------------------------------
C [q]
using gpi

→ ProofInfoLe
    (ExGen := list_to_set
                (evar_fresh_seq
                   (free_evars (pcPattern C)
                    ∪ free_evars p ∪ free_evars q
                    ∪ {[pcEvar C]})
                   (maximal_exists_depth_of_evar_in_pattern
                      (pcEvar C) (pcPattern C))),
     SVSubst := list_to_set
                  (svar_fresh_seq
                     (free_svars (pcPattern C)
                      ∪ free_svars p ∪ free_svars q)
                     (maximal_mu_depth_of_evar_in_pattern
                        (pcEvar C) (pcPattern C))),
     KT := if
            decide
              (0 =
               maximal_mu_depth_of_evar_in_pattern
                 (pcEvar C) (pcPattern C))
           then false
           else true,
     FP := gset_to_coGset
             (frames_on_the_way_to_hole'
                (free_evars (pcPattern C)
                 ∪ free_evars p ∪ free_evars q
                 ∪ {[pcEvar C]})
                (free_svars (pcPattern C)
                 ∪ free_svars p ∪ free_svars q)
                (pcEvar C) (pcPattern C) p q wfC
                (extract_wfp Γ p q gpi Hpiffq)
                (extract_wfq Γ p q gpi Hpiffq))) gpi
  → Γ Ⱶ
  l--------------------------------------
  C [p]
  using gpi
  
  intros H pile.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ Ⱶ
l--------------------------------------
C [q]
using gpi

pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
Γ Ⱶ
l--------------------------------------
C [p]
using gpi

  unfold of_MLGoal in *.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: well_formed
  (mlConclusion
     (Γ Ⱶ
      l--------------------------------------
      C [q]
      using gpi
      ))
→ wf
    (patterns_of
       (mlHypotheses
          (Γ Ⱶ
           l--------------------------------------
           C [q]
           using gpi
           )))
  → mlTheory
      (Γ Ⱶ
       l--------------------------------------
       C [q]
       using gpi
       )
    ⊢i foldr patt_imp
         (mlConclusion
            (Γ Ⱶ
             l--------------------------------------
             C [q]
             using gpi
             ))
         (patterns_of
            (mlHypotheses
               (Γ Ⱶ
                l--------------------------------------
                C [q]
                using gpi
                )))
    using mlInfo
            (Γ Ⱶ
             l--------------------------------------
             C [q]
             using gpi
             )
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
well_formed
  (mlConclusion
     (Γ Ⱶ
      l--------------------------------------
      C [p]
      using gpi
      ))
→ wf
    (patterns_of
       (mlHypotheses
          (Γ Ⱶ
           l--------------------------------------
           C [p]
           using gpi
           )))
  → mlTheory
      (Γ Ⱶ
       l--------------------------------------
       C [p]
       using gpi
       )
    ⊢i foldr patt_imp
         (mlConclusion
            (Γ Ⱶ
             l--------------------------------------
             C [p]
             using gpi
             ))
         (patterns_of
            (mlHypotheses
               (Γ Ⱶ
                l--------------------------------------
                C [p]
                using gpi
                )))
    using mlInfo
            (Γ Ⱶ
             l--------------------------------------
             C [p]
             using gpi
             ) simpl in *.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: well_formed (C [q])
→ wf (patterns_of l)
  → Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
    using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
well_formed (C [p])
→ wf (patterns_of l)
  → Γ ⊢i foldr patt_imp (C [p]) (patterns_of l)
    using gpi
  intros wfcp wfl.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: well_formed (C [q])
→ wf (patterns_of l)
  → Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
    using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i foldr patt_imp (C [p]) (patterns_of l) using gpi
  feed specialize H.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: well_formed (C [q])
→ wf (patterns_of l)
  → Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
    using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
well_formed (C [q])
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: well_formed (C [q])
→ wf (patterns_of l)
  → Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
    using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
wf (patterns_of l)
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i foldr patt_imp (C [p]) (patterns_of l) using gpi
  {
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: well_formed (C [q])
→ wf (patterns_of l)
  → Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
    using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
well_formed (C [q]) abstract (
      pose proof (Hwfiff := proved_impl_wf _ _ (proj1_sig Hpiffq));
      unfold emplace;
      apply well_formed_free_evar_subst_0;[wf_auto2|];
      fold (PC_wf C);
      eapply wf_emplaced_impl_wf_context;
      apply wfcp
    ).
  }
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: well_formed (C [q])
→ wf (patterns_of l)
  → Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
    using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
wf (patterns_of l)
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i foldr patt_imp (C [p]) (patterns_of l) using gpi
  {
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: well_formed (C [q])
→ wf (patterns_of l)
  → Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
    using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
wf (patterns_of l) exact wfl. }
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i foldr patt_imp (C [p]) (patterns_of l) using gpi

  eapply MP.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i ?ϕ₁ using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i ?ϕ₁ ---> foldr patt_imp (C [p]) (patterns_of l)
using gpi
  2: apply pf_iff_proj2.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i ?ϕ₁ using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
well_formed (foldr patt_imp (C [p]) (patterns_of l))
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
well_formed ?ϕ₁
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i foldr patt_imp (C [p]) (patterns_of l) <---> ?ϕ₁
using gpi
  2: abstract (wf_auto2).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i ?ϕ₁ using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
well_formed ?ϕ₁
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i foldr patt_imp (C [p]) (patterns_of l) <---> ?ϕ₁
using gpi
  3: eapply prf_equiv_congruence_iter.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i foldr patt_imp (C [?q]) (patterns_of l) using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
well_formed (foldr patt_imp (C [?q]) (patterns_of l))
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars ?q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p ∪ 
                    free_svars ?q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C) ∪ free_evars p
               ∪ free_evars ?q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C) ∪ free_svars p
               ∪ free_svars ?q) 
              (pcEvar C) 
              (pcPattern C) p 
              ?q ?wfC ?wfp 
              ?wfq)) gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
wf (patterns_of l)
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i p <---> ?q using gpi
  5: apply Hpiffq.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i foldr patt_imp (C [q]) (patterns_of l) using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
well_formed (foldr patt_imp (C [q]) (patterns_of l))
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p ∪ 
                    free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C) ∪ free_evars p
               ∪ free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C) ∪ free_svars p
               ∪ free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q 
              ?wfC ?wfp 
              ?wfq)) gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
wf (patterns_of l)
  4: assumption.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
Γ ⊢i foldr patt_imp (C [q]) (patterns_of l) using gpi
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
well_formed (foldr patt_imp (C [q]) (patterns_of l))
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p ∪ 
                    free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C) ∪ free_evars p
               ∪ free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C) ∪ free_svars p
               ∪ free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q 
              ?wfC ?wfp 
              ?wfq)) gpi
  1: apply H.
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
well_formed (foldr patt_imp (C [q]) (patterns_of l))
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p ∪ 
                    free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C) ∪ free_evars p
               ∪ free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C) ∪ free_svars p
               ∪ free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q 
              ?wfC ?wfp 
              ?wfq)) gpi
  1: {
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
well_formed (foldr patt_imp (C [q]) (patterns_of l))
    pose proof (proved_impl_wf _ _ (proj1_sig H)).
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
H0: well_formed
  (foldr patt_imp (C [q]) (patterns_of l))
well_formed (foldr patt_imp (C [q]) (patterns_of l))
    wf_auto2.
  }
Σ: Signature
Γ: Theory
p, q: Pattern
C: PatternCtx
l: hypotheses
gpi: ProofInfo
wfC: PC_wf C
Hpiffq: Γ ⊢i p <---> q using gpi
H: Γ ⊢i foldr patt_imp (C [q]) (patterns_of l)
using gpi
pile: ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ 
                  free_evars q ∪ {[
                  pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) 
                    (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p
                    ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) 
                      (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) 
               (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C)
               ∪ free_evars p ∪ 
               free_evars q ∪ {[
               pcEvar C]})
              (free_svars (pcPattern C)
               ∪ free_svars p ∪ 
               free_svars q) 
              (pcEvar C) 
              (pcPattern C) p q wfC
              (extract_wfp Γ p q gpi Hpiffq)
              (extract_wfq Γ p q gpi Hpiffq)))
  gpi
wfcp: well_formed (C [p])
wfl: wf (patterns_of l)
ProofInfoLe
  (ExGen := list_to_set
              (evar_fresh_seq
                 (free_evars (pcPattern C)
                  ∪ free_evars p ∪ free_evars q
                  ∪ {[pcEvar C]})
                 (maximal_exists_depth_of_evar_in_pattern
                    (pcEvar C) (pcPattern C))),
   SVSubst := list_to_set
                (svar_fresh_seq
                   (free_svars (pcPattern C)
                    ∪ free_svars p ∪ free_svars q)
                   (maximal_mu_depth_of_evar_in_pattern
                      (pcEvar C) (pcPattern C))),
   KT := if
          decide
            (0 =
             maximal_mu_depth_of_evar_in_pattern
               (pcEvar C) (pcPattern C))
         then false
         else true,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              (free_evars (pcPattern C) ∪ free_evars p
               ∪ free_evars q ∪ {[pcEvar C]})
              (free_svars (pcPattern C) ∪ free_svars p
               ∪ free_svars q) (pcEvar C)
              (pcPattern C) p q ?wfC ?wfp ?wfq)) gpi
  exact pile.
Defined.




Ltac2 mutable ml_debug_rewrite := false.

(* Calls [cont] for every subpattern [a] of pattern [phi], giving the match context as an argument *)
Ltac2 for_each_match := fun (a : constr) (phi : constr) (cont : Pattern.context -> unit) =>
  try (
      if ml_debug_rewrite then
           Message.print (
               Message.concat
                 (Message.of_string "Trying to match ")
                 (Message.of_constr a)
             )
        else ();
      match! phi with
      | context ctx [ ?x ]
        => if ml_debug_rewrite then
             Message.print (
                 Message.concat
                   (Message.of_string " against ")
                   (Message.of_constr x)
               )
           else ();
           (if Constr.equal x a then
              if ml_debug_rewrite then
                Message.print (Message.of_string "Success.")
              else () ;
              cont ctx
            else ());
           fail (* backtrack *)
      end
    ); ().

(* Calls [cont] for [n]th subpatern [a] of pattern [phi]. *)
Ltac2 for_nth_match :=
  fun (n : int) (a : constr) (phi : constr) (cont : Pattern.context -> unit) =>
    if ml_debug_rewrite then
      Message.print (Message.of_string "for_nth_match")
    else () ;
    let curr : int ref := {contents := 0} in
    let found : bool ref := {contents := false} in
    for_each_match a phi
    (fun ctx =>
      if (found.(contents))
      then ()
      else
        curr.(contents) := Int.add 1 (curr.(contents)) ;
        if (Int.equal (curr.(contents)) n) then
          cont ctx
        else ()
    )
.

Local Ltac reduce_free_evar_subst_step_2 star :=
      lazymatch goal with
      | [ |- context ctx [?p^[[evar: star ↦ ?q]] ] ]
        =>
          progress rewrite -> (@free_evar_subst_no_occurrence _ star p q) by (
            apply count_evar_occurrences_0;
            subst star;
            eapply evar_is_fresh_in_richer';
            [|apply set_evar_fresh_is_fresh'];
            simpl; clear; set_solver
          )
      end.

Local Ltac reduce_free_evar_subst_2 star :=
  (* unfold free_evar_subst; *)
  repeat (reduce_free_evar_subst_step_2 star).

Local Tactic Notation "solve_fresh_contradictions_2'" constr(star) constr(x) constr(h) :=
  let hcontra := fresh "Hcontra" in
  assert (hcontra: x <> star) by (subst star; unfold fresh_evar,evar_fresh_s; try clear h; simpl; solve_fresh_neq);
  rewrite -> h in hcontra;
  contradiction.

Local Ltac solve_fresh_contradictions_2 star :=
  unfold fresh_evar; simpl;
  match goal with
  | h: ?x = star |- _ =>
    let hprime := fresh "hprime" in
    pose proof (hprime := eq_sym h);
    solve_fresh_contradictions_2' star x hprime
  | h: star = ?x |- _
    => solve_fresh_contradictions_2' star x h
  end.

Local Ltac clear_obvious_equalities_2 :=
  repeat (
      match goal with
      | [ h: ?x = ?x |- _ ] => clear h
      end
    ).


Ltac simplify_emplace_2 star :=
  unfold emplace;
  simpl;
  (* unfold free_evar_subst; *)
  simpl;
  repeat break_match_goal;
  clear_obvious_equalities_2; try contradiction;
  try (solve_fresh_contradictions_2 star);
  (* repeat (rewrite nest_ex_aux_0); *)
  reduce_free_evar_subst_2 star.

(* Returns [n]th matching logic context [C] (of type [PatternCtx]) such that
   [emplace C a = phi].
 *)

 
 Ltac simplify_pile_side_condition_helper star :=
  subst star;
  unfold fresh_evar,evar_fresh_s;
  eapply evar_is_fresh_in_richer';
  [|apply set_evar_fresh_is_fresh'];
  clear; simpl; set_solver.

 Ltac simplify_pile_side_condition star :=
  try apply pile_any;
  cbn;
  simplify_emplace_2 star;
  repeat (rewrite (mmdoeip_notin, medoeip_notin);
  [(simplify_pile_side_condition_helper star)|]);
  simpl;
  repeat (
    lazymatch goal with
    | [H: context [maximal_mu_depth_of_evar_in_pattern' _ _ _] |- _ ]
      => rewrite mmdoeip_notin in H;
         [(simplify_pile_side_condition_helper star)|]
    | [H: context [maximal_exists_depth_of_evar_in_pattern' _ _ _] |- _ ]
      => rewrite medoeip_notin in H;
         [(simplify_pile_side_condition_helper star)|]
    end
  );
  simpl in *;
  try lia;
  try apply pile_any.

Ltac2 Type HeatResult := {
  star_ident : ident ;
  star_eq : ident ;
  pc : constr ;
  ctx : Pattern.context ;
  ctx_pat : constr ;
  equality : ident ;
}.

Ltac2 heat :=
  fun (n : int) (a : constr) (phi : constr) : HeatResult =>
    let found : (Pattern.context option) ref := { contents := None } in
     for_nth_match n a phi
     (fun ctx =>
        found.(contents) := Some ctx; ()
     );
    match found.(contents) with
    | None => Control.backtrack_tactic_failure "Cannot heat"
    | Some ctx
      => (
         let fr := constr:(fresh_evar $phi) in
         let star_ident := Fresh.in_goal ident:(star) in
         let star_eq := Fresh.in_goal ident:(star_eq) in
         (*set ($star_ident := $fr);*)
         remember $fr as $star_ident eqn:star_eq;
         let star_hyp := Control.hyp star_ident in
         let ctxpat := Pattern.instantiate ctx constr:(patt_free_evar $star_hyp) in
         let pc := constr:((@Build_PatternCtx _ $star_hyp $ctxpat)) in
         let heq1 := Fresh.in_goal ident:(heq1) in
         assert(heq1 : ($phi = (@emplace _ $pc $a))) 
         > [ abstract(
             (ltac1:(star |- simplify_emplace_2 star) (Ltac1.of_ident star_ident);
             reflexivity
             ))
           | ()
           ];
         { star_ident := star_ident; star_eq := star_eq; pc := pc; ctx := ctx; ctx_pat := ctxpat; equality := heq1 }
         )
    end
.

Ltac2 mlRewrite (hiff : constr) (atn : int) :=
  let thiff := Constr.type hiff in
  (* we have to unfold [derives] otherwise this might not match *)
  lazy_match! (eval unfold ProofSystem.derives in $thiff) with
  | _ ⊢i (?a <---> ?a') using _
    =>
    unfold AnyReasoning;
    lazy_match! goal with
    | [ |- of_MLGoal (@mkMLGoal ?sgm ?g ?l ?p ( ?gpi))]
      => let hr : HeatResult := heat atn a p in
         if ml_debug_rewrite then
           Message.print (Message.of_constr (hr.(ctx_pat)))
         else () ;
         let heq := Control.hyp (hr.(equality)) in
         let pc := (hr.(pc)) in
         eapply (@cast_proof_ml_goal _ $g) >
           [ rewrite $heq; reflexivity | ()];
         Std.clear [hr.(equality)];
         let wfC := Fresh.in_goal ident:(wfC) in
         assert (wfC : PC_wf $pc = true) > [ ltac1:(unfold PC_wf; simpl; wf_auto2); Control.shelve () | ()] ;
         let wfCpf := Control.hyp wfC in
         apply (@MLGoal_rewriteIff $sgm $g _ _ $pc $l $gpi $wfCpf $hiff)  >
         [
         (lazy_match! goal with
         | [ |- of_MLGoal (@mkMLGoal ?sgm ?g ?l ?p _)]
           =>
             let heq2 := Fresh.in_goal ident:(heq2) in
             let plugged := Pattern.instantiate (hr.(ctx)) a' in
             assert(heq2: ($p = $plugged))
             > [
                 abstract (ltac1:(star |- simplify_emplace_2 star) (Ltac1.of_ident (hr.(star_ident)));
                 reflexivity
                 )
               | ()
               ];
             let heq2_pf := Control.hyp heq2 in
             eapply (@cast_proof_ml_goal _ $g) >
               [ rewrite $heq2_pf; reflexivity | ()];
             Std.clear [wfC; heq2 ; (hr.(star_ident)); (hr.(star_eq))]
         end)
         | (ltac1:(star |- simplify_pile_side_condition star) (Ltac1.of_ident (hr.(star_ident))))
         ]
    end
  end.

Ltac2 rec constr_to_int (x : constr) : int :=
  match! x with
  | 0 => 0
  | (S ?x') => Int.add 1 (constr_to_int x')
  end.


Tactic Notation "mlRewrite" constr(Hiff) "at" constr(atn) :=
  (let ff := ltac2:(hiff atn |-
                      mlRewrite
                        (Option.get (Ltac1.to_constr(hiff)))
                        (constr_to_int (Option.get (Ltac1.to_constr(atn))))
                   ) in
   ff Hiff atn).

Lemma pf_iff_equiv_sym_nowf {Σ : Signature} Γ A B i :
  Γ ⊢i (A <---> B) using i ->
  Γ ⊢i (B <---> A) using i.
Σ: Signature
Γ: Theory
A, B: Pattern
i: ProofInfo
Γ ⊢i A <---> B using i → Γ ⊢i B <---> A using i
Proof.
Σ: Signature
Γ: Theory
A, B: Pattern
i: ProofInfo
Γ ⊢i A <---> B using i → Γ ⊢i B <---> A using i
  intros H.
Σ: Signature
Γ: Theory
A, B: Pattern
i: ProofInfo
H: Γ ⊢i A <---> B using i
Γ ⊢i B <---> A using i
  pose proof (wfp := proved_impl_wf _ _ (proj1_sig H)).
Σ: Signature
Γ: Theory
A, B: Pattern
i: ProofInfo
H: Γ ⊢i A <---> B using i
wfp: well_formed (A <---> B)
Γ ⊢i B <---> A using i
  assert (well_formed A) by wf_auto2.
Σ: Signature
Γ: Theory
A, B: Pattern
i: ProofInfo
H: Γ ⊢i A <---> B using i
wfp: well_formed (A <---> B)
H0: well_formed A
Γ ⊢i B <---> A using i
  assert (well_formed B) by wf_auto2.
Σ: Signature
Γ: Theory
A, B: Pattern
i: ProofInfo
H: Γ ⊢i A <---> B using i
wfp: well_formed (A <---> B)
H0: well_formed A
H1: well_formed B
Γ ⊢i B <---> A using i
  apply pf_iff_equiv_sym; assumption.
Defined.

Tactic Notation "mlRewrite" "->" constr(Hiff) "at" constr(atn) :=
  mlRewrite Hiff at atn.

Tactic Notation "mlRewrite" "<-" constr(Hiff) "at" constr(atn) :=
  mlRewrite (@pf_iff_equiv_sym_nowf _ _ _ _ _ Hiff) at atn.


Local Example ex_prf_rewrite_equiv_2 {Σ : Signature} Γ a a' b x:
  well_formed a ->
  well_formed a' ->
  well_formed b ->
  Γ ⊢ a <---> a' ->
  Γ ⊢i (a $ a $ b $ a ---> (patt_free_evar x)) <---> (a $ a' $ b $ a' ---> (patt_free_evar x))
  using AnyReasoning.
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
well_formed a
→ well_formed a'
  → well_formed b
    → Γ ⊢ a <---> a'
      → Γ
        ⊢i (a $ a $ b $ a ---> patt_free_evar x) <--->
           (a $ a' $ b $ a' ---> patt_free_evar x)
        using AnyReasoning
Proof.
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
well_formed a
→ well_formed a'
  → well_formed b
    → Γ ⊢ a <---> a'
      → Γ
        ⊢i (a $ a $ b $ a ---> patt_free_evar x) <--->
           (a $ a' $ b $ a' ---> patt_free_evar x)
        using AnyReasoning
  intros wfa wfa' wfb Hiff.
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
wfa: well_formed a
wfa': well_formed a'
wfb: well_formed b
Hiff: Γ ⊢ a <---> a'
Γ
⊢i (a $ a $ b $ a ---> patt_free_evar x) <--->
   (a $ a' $ b $ a' ---> patt_free_evar x)
using AnyReasoning
  toMLGoal.
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
wfa: well_formed a
wfa': well_formed a'
wfb: well_formed b
Hiff: Γ ⊢ a <---> a'
well_formed
  ((a $ a $ b $ a ---> patt_free_evar x) <--->
   (a $ a' $ b $ a' ---> patt_free_evar x))
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
wfa: well_formed a
wfa': well_formed a'
wfb: well_formed b
Hiff: Γ ⊢ a <---> a'
Γ Ⱶ
 (a $ a $ b $ a ---> patt_free_evar x) <--->
 (a $ a' $ b $ a' ---> patt_free_evar x)

  {
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
wfa: well_formed a
wfa': well_formed a'
wfb: well_formed b
Hiff: Γ ⊢ a <---> a'
well_formed
  ((a $ a $ b $ a ---> patt_free_evar x) <--->
   (a $ a' $ b $ a' ---> patt_free_evar x)) abstract(wf_auto2). }
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
wfa: well_formed a
wfa': well_formed a'
wfb: well_formed b
Hiff: Γ ⊢ a <---> a'
Γ Ⱶ
 (a $ a $ b $ a ---> patt_free_evar x) <--->
 (a $ a' $ b $ a' ---> patt_free_evar x)

  mlRewrite Hiff at 2.
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
wfa: well_formed a
wfa': well_formed a'
wfb: well_formed b
Hiff: Γ ⊢ a <---> a'
Γ Ⱶ
(a $ a' $ b $ a ---> patt_free_evar x) <--->
(a $ a' $ b $ a' ---> patt_free_evar x)
using (ExGen := ⊤, SVSubst := ⊤, KT := true, FP := ⊤)

  mlRewrite <- Hiff at 3.
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
wfa: well_formed a
wfa': well_formed a'
wfb: well_formed b
Hiff: Γ ⊢ a <---> a'
Γ Ⱶ
(a $ a' $ b $ a ---> patt_free_evar x) <--->
(a $ a' $ b $ a ---> patt_free_evar x)
using (ExGen := ⊤, SVSubst := ⊤, KT := true, FP := ⊤)

  fromMLGoal.
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
wfa: well_formed a
wfa': well_formed a'
wfb: well_formed b
Hiff: Γ ⊢ a <---> a'
Γ
⊢i (a $ a' $ b $ a ---> patt_free_evar x) <--->
   (a $ a' $ b $ a ---> patt_free_evar x)
using (ExGen := ⊤, SVSubst := ⊤, KT := true, FP := ⊤)
  useBasicReasoning.
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
wfa: well_formed a
wfa': well_formed a'
wfb: well_formed b
Hiff: Γ ⊢ a <---> a'
Γ
⊢i (a $ a' $ b $ a ---> patt_free_evar x) <--->
   (a $ a' $ b $ a ---> patt_free_evar x)
using BasicReasoning
  apply pf_iff_equiv_refl.
Σ: Signature
Γ: Theory
a, a', b: Pattern
x: evar
wfa: well_formed a
wfa': well_formed a'
wfb: well_formed b
Hiff: Γ ⊢ a <---> a'
well_formed (a $ a' $ b $ a ---> patt_free_evar x) abstract(wf_auto2).
Defined.



(* TODO: de-duplicate the code *)
#[local]
Ltac convertToNNF_rewrite_pat Ctx p i :=
  lazymatch p with
    | (! ! ?x) =>
        let H' := fresh "H" in
        pose proof (@not_not_eq _ Ctx x ltac:(wf_auto2)) as H';
        apply (@useBasicReasoning _ _ _ i) in H';
        repeat (mlRewrite H' at 1);
        try clear H';
        convertToNNF_rewrite_pat Ctx x i
    | patt_not (patt_and ?x ?y) =>
        let H' := fresh "H" in
        pose proof (@deMorgan_nand _ Ctx x y ltac:(wf_auto2) ltac:(wf_auto2)) as H';
        apply (@useBasicReasoning _ _ _ i) in H';
        repeat (mlRewrite H' at 1);
        try clear H';
        convertToNNF_rewrite_pat Ctx (!x or !y) i
    | patt_not (patt_or ?x ?y) =>
        let H' := fresh "H" in
        pose proof (@deMorgan_nor _ Ctx x y ltac:(wf_auto2) ltac:(wf_auto2)) as H';
        apply (@useBasicReasoning _ _ _ i) in H';
        repeat (mlRewrite H' at 1);
        try clear H';
        convertToNNF_rewrite_pat Ctx (!x and !y) i
    | patt_not (?x ---> ?y) =>
        let H' := fresh "H" in
        pose proof (@nimpl_eq_and _ Ctx x y ltac:(wf_auto2) ltac:(wf_auto2)) as H';
        apply (@useBasicReasoning _ _ _ i) in H';
        repeat (mlRewrite H' at 1);
        try clear H';
        convertToNNF_rewrite_pat Ctx (x and !y) i
    | (?x ---> ?y) =>
        let H' := fresh "H" in
        pose proof (@impl_eq_or _ Ctx x y ltac:(wf_auto2) ltac:(wf_auto2)) as H';
        apply (@useBasicReasoning _ _ _ i) in H';
        repeat (mlRewrite H' at 1);
        try clear H';
        convertToNNF_rewrite_pat Ctx (!x or y) i
    | patt_and ?x ?y => convertToNNF_rewrite_pat Ctx x i; convertToNNF_rewrite_pat Ctx y i
    | patt_or ?x ?y => convertToNNF_rewrite_pat Ctx x i; convertToNNF_rewrite_pat Ctx y i
    | _ => idtac
  end.

#[local]
Ltac toNNF := 
  repeat mlRevertLast;
  match goal with
    | [ |- @of_MLGoal ?Sgm (@mkMLGoal ?Sgm ?Ctx ?ll ?g ?i) ] 
      =>
        mlApplyMetaRaw (@useBasicReasoning _ _ _ i (@not_not_elim Sgm Ctx g ltac:(wf_auto2)));
        convertToNNF_rewrite_pat Ctx (!g) i
  end.

#[local] Example test_toNNF {Σ : Signature} Γ a b :
  well_formed a ->
  well_formed b ->
  Γ ⊢i ( (b and (a or b) and !b and ( a or a) and a) ---> ⊥)
  using BasicReasoning.
Σ: Signature
Γ: Theory
a, b: Pattern
well_formed a
→ well_formed b
  → Γ
    ⊢i b and (a or b) and ! b and (a or a) and a --->
       ⊥ using BasicReasoning
Proof.
Σ: Signature
Γ: Theory
a, b: Pattern
well_formed a
→ well_formed b
  → Γ
    ⊢i b and (a or b) and ! b and (a or a) and a --->
       ⊥ using BasicReasoning
  intros wfa wfb.
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ ⊢i b and (a or b) and ! b and (a or a) and a ---> ⊥
using BasicReasoning
  toMLGoal.
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
well_formed
  (b and (a or b) and ! b and (a or a) and a ---> ⊥)
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ Ⱶ
b and (a or b) and ! b and (a or a) and a ---> ⊥
using BasicReasoning

  {
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
well_formed
  (b and (a or b) and ! b and (a or a) and a ---> ⊥) wf_auto2. }
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ Ⱶ
b and (a or b) and ! b and (a or a) and a ---> ⊥
using BasicReasoning

  toNNF.
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ Ⱶ
! ((b and (a or b) and ! b and (a or a) and a) and ! ⊥)
using BasicReasoning
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
H: Γ
⊢i ! (b and (a or b) and ! b and (a or a) and a --->
      ⊥) <--->
   (b and (a or b) and ! b and (a or a) and a)
   and ! ⊥ using BasicReasoning
star: evar
star_eq: star =
fresh_evar
  (! ! (b
        and (a or b)
            and ! b and (a or a) and a --->
        ⊥))
wfC: PC_wf
  {|
    pcEvar := star;
    pcPattern := ! patt_free_evar star
  |} = true
e0: 0 = 0
ProofInfoLe
  (ExGen := ∅, SVSubst := ∅, KT := false,
   FP := gset_to_coGset
           (frames_on_the_way_to_hole'
              ({[star]} ∪ ∅
               ∪ (free_evars b ∪ ∅ ∪ ∅
                  ∪ (free_evars a ∪ ∅ ∪ free_evars b
                     ∪ ∅ ∪ ∅
                     ∪ (free_evars b ∪ ∅ ∪ ∅ ∪ ∅
                        ∪ (... ∪ ... ∪ ∅ ∪ ∅) ∪ ∅ ∪ ∅)
                     ∪ ∅ ∪ ∅) ∪ ∅ ∪ ∅ ∪ ∅)
               ∪ (free_evars b ∪ ∅ ∪ ∅
                  ∪ (free_evars a ∪ ∅ ∪ free_evars b
                     ∪ ∅ ∪ ∅
                     ∪ (... ∪ ∅ ∪ ∅ ∪ ∅
                        ∪ (... ∪ ∅ ∪ ∅) ∪ ∅ ∪ ∅) ∪ ∅
                     ∪ ∅) ∪ ∅ ∪ ∅ ∪ ∅ ∪ (∅ ∪ ∅ ∪ ∅)
                  ∪ ∅) ∪ {[star]})
              (∅ ∪ ∅
               ∪ (free_svars b ∪ ∅ ∪ ∅
                  ∪ (free_svars a ∪ ∅ ∪ free_svars b
                     ∪ ∅ ∪ ∅
                     ∪ (free_svars b ∪ ∅ ∪ ∅ ∪ ∅
                        ∪ (... ∪ ∅ ∪ (...) ∪ ∅ ∪ ∅)
                        ∪ ∅ ∪ ∅) ∪ ∅ ∪ ∅) ∪ ∅ ∪ ∅ ∪ ∅)
               ∪ (free_svars b ∪ ∅ ∪ ∅
                  ∪ (free_svars a ∪ ∅ ∪ free_svars b
                     ∪ ∅ ∪ ∅
                     ∪ (free_svars b ∪ ∅ ∪ ∅ ∪ ∅
                        ∪ (... ∪ ... ∪ ∅ ∪ ∅) ∪ ∅ ∪ ∅)
                     ∪ ∅ ∪ ∅) ∪ ∅ ∪ ∅ ∪ ∅
                  ∪ (∅ ∪ ∅ ∪ ∅) ∪ ∅)) star
              (! patt_free_evar star)
              (! (b
                  and (a or b)
                      and ! b and (a or a) and a --->
                  ⊥))
              ((b
                and (a or b)
                    and ! b and (a or a) and a)
               and ! ⊥) wfC
              (extract_wfp Γ
                 (! (b
                     and (a or b)
                         and ! b and (a or a) and a --->
                     ⊥))
                 ((b
                   and (a or b)
                       and ! b and (a or a) and a)
                  and ! ⊥) BasicReasoning H)
              (extract_wfq Γ
                 (! (b
                     and (a or b)
                         and ! b and (a or a) and a --->
                     ⊥))
                 ((b
                   and (a or b)
                       and ! b and (a or a) and a)
                  and ! ⊥) BasicReasoning H)))
  BasicReasoning
Abort.

#[local]
Ltac rfindContradictionTo a ll k :=
  match ll with
    | ((mkNH _ ?name (! a)) :: ?m) =>
        mlApply name; mlExactn k
    | ((mkNH _ _ _) :: ?m) => 
        rfindContradictionTo a m k
    | _ => fail
  end.

#[local]
Ltac findContradiction l k:=
    match l with
       | ((mkNH _ _ ?a) :: ?m) => 
             match goal with
                | [ |- @of_MLGoal ?Sgm (@mkMLGoal ?Sgm ?Ctx ?ll ?g ?i) ] 
                  =>
                     try rfindContradictionTo a ll k;
                     let kk := eval compute in ( k + 1 ) in
                     (findContradiction m kk)
             end
       | _ => fail
    end.

#[local]
Ltac findContradiction_start :=
  match goal with
    | [ |- @of_MLGoal ?Sgm (@mkMLGoal ?Sgm ?Ctx ?l ?g ?i) ] 
      =>
        match goal with
          | [ |- @of_MLGoal ?Sgm (@mkMLGoal ?Sgm ?Ctx ?l ?g ?i) ] 
            =>
              findContradiction l 0
        end
  end.

#[local]
Ltac breakHyps l :=
  match l with
  | ((mkNH _ ?name (?x and ?y)) :: ?m) => 
      mlDestructAnd name
  | ((mkNH _ ?name (?x or ?y)) :: ?m) => 
      mlDestructOr name
  | ((mkNH _ ?name ?x) :: ?m)  =>
      breakHyps m
  end.

#[local]
Ltac mlTautoBreak := repeat match goal with
| [ |- @of_MLGoal ?Sgm (@mkMLGoal ?Sgm ?Ctx ?l ?g ?i) ] 
  =>
    lazymatch g with
      | (⊥) =>
              breakHyps l
      | _ => mlApplyMetaRaw (@useBasicReasoning _ _ _ i (@bot_elim _ _ g _))
    end
end.

Ltac try_solve_pile2 fallthrough :=
  lazymatch goal with
  | [ |- ProofInfoLe _ _] => try apply pile_refl; try_solve_pile; fallthrough
  | _ => idtac
  end.

#[global]
Ltac mlTauto :=
  unshelve(
    try (
      toNNF; (try_solve_pile2 shelve);
      repeat mlIntro;
      mlTautoBreak;
      findContradiction_start
    )
  )
.

#[local]
Example conj_right {Σ : Signature} Γ a b:
  well_formed a ->
  well_formed b ->
  Γ ⊢i ( (b and (a or b) and !b and ( a or a) and a) ---> ⊥)
  using AnyReasoning.
Σ: Signature
Γ: Theory
a, b: Pattern
well_formed a
→ well_formed b
  → Γ
    ⊢i b and (a or b) and ! b and (a or a) and a --->
       ⊥ using AnyReasoning
Proof.
Σ: Signature
Γ: Theory
a, b: Pattern
well_formed a
→ well_formed b
  → Γ
    ⊢i b and (a or b) and ! b and (a or a) and a --->
       ⊥ using AnyReasoning
  intros wfa wfb.
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ ⊢i b and (a or b) and ! b and (a or a) and a ---> ⊥
using AnyReasoning
  toMLGoal.
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
well_formed
  (b and (a or b) and ! b and (a or a) and a ---> ⊥)
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ Ⱶ
 b and (a or b) and ! b and (a or a) and a ---> ⊥

  {
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
well_formed
  (b and (a or b) and ! b and (a or a) and a ---> ⊥) wf_auto2. }
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ Ⱶ
 b and (a or b) and ! b and (a or a) and a ---> ⊥

  (* TODO: fail loudly if there is something else than AnyReasoning *)
  mlTauto.
Defined.

#[local]
Example condtradict_taut_2 {Σ : Signature} Γ a b:
  well_formed a ->
  well_formed b ->
  Γ ⊢i (a ---> ((! a) ---> b))
  using AnyReasoning.
Σ: Signature
Γ: Theory
a, b: Pattern
well_formed a
→ well_formed b
  → Γ ⊢i a ---> ! a ---> b using AnyReasoning
Proof.
Σ: Signature
Γ: Theory
a, b: Pattern
well_formed a
→ well_formed b
  → Γ ⊢i a ---> ! a ---> b using AnyReasoning
  intros wfa wfb.
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ ⊢i a ---> ! a ---> b using AnyReasoning
  toMLGoal.
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
well_formed (a ---> ! a ---> b)
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ Ⱶ
 a ---> ! a ---> b

  {
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
well_formed (a ---> ! a ---> b) wf_auto2. }
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ Ⱶ
 a ---> ! a ---> b

  mlTauto.
Defined.

#[local]
Example taut {Σ : Signature} Γ a b c:
  well_formed a ->
  well_formed b ->
  well_formed c ->
  Γ ⊢i ((a ---> b) ---> ((b ---> c) ---> ((a or b)---> c)))
  using AnyReasoning.
Σ: Signature
Γ: Theory
a, b, c: Pattern
well_formed a
→ well_formed b
  → well_formed c
    → Γ
      ⊢i (a ---> b) ---> (b ---> c) ---> a or b ---> c
      using AnyReasoning
Proof.
Σ: Signature
Γ: Theory
a, b, c: Pattern
well_formed a
→ well_formed b
  → well_formed c
    → Γ
      ⊢i (a ---> b) ---> (b ---> c) ---> a or b ---> c
      using AnyReasoning
  intros wfa wfb wfc.
Σ: Signature
Γ: Theory
a, b, c: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
Γ ⊢i (a ---> b) ---> (b ---> c) ---> a or b ---> c
using AnyReasoning
  toMLGoal.
Σ: Signature
Γ: Theory
a, b, c: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
well_formed
  ((a ---> b) ---> (b ---> c) ---> a or b ---> c)
Σ: Signature
Γ: Theory
a, b, c: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
Γ Ⱶ
 (a ---> b) ---> (b ---> c) ---> a or b ---> c

  {
Σ: Signature
Γ: Theory
a, b, c: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
well_formed
  ((a ---> b) ---> (b ---> c) ---> a or b ---> c) wf_auto2. }
Σ: Signature
Γ: Theory
a, b, c: Pattern
wfa: well_formed a
wfb: well_formed b
wfc: well_formed c
Γ Ⱶ
 (a ---> b) ---> (b ---> c) ---> a or b ---> c

  mlTauto. (* Slow *)
Defined.

#[local]
Example condtradict_taut_1 {Σ : Signature} Γ a:
  well_formed a ->
  Γ ⊢i !(a and !a)
  using AnyReasoning.
Σ: Signature
Γ: Theory
a: Pattern
well_formed a → Γ ⊢i ! (a and ! a) using AnyReasoning
Proof.
Σ: Signature
Γ: Theory
a: Pattern
well_formed a → Γ ⊢i ! (a and ! a) using AnyReasoning
  intros wfa.
Σ: Signature
Γ: Theory
a: Pattern
wfa: well_formed a
Γ ⊢i ! (a and ! a) using AnyReasoning
  toMLGoal.
Σ: Signature
Γ: Theory
a: Pattern
wfa: well_formed a
well_formed (! (a and ! a))
Σ: Signature
Γ: Theory
a: Pattern
wfa: well_formed a
Γ Ⱶ
 ! (a and ! a)

  {
Σ: Signature
Γ: Theory
a: Pattern
wfa: well_formed a
well_formed (! (a and ! a)) wf_auto2. }
Σ: Signature
Γ: Theory
a: Pattern
wfa: well_formed a
Γ Ⱶ
 ! (a and ! a)

  mlTauto.
Defined.

#[local]
Example notnot_taut_1 {Σ : Signature} Γ a:
  well_formed a ->
  Γ ⊢i (! ! a ---> a)
  using AnyReasoning.
Σ: Signature
Γ: Theory
a: Pattern
well_formed a → Γ ⊢i ! ! a ---> a using AnyReasoning
Proof.
Σ: Signature
Γ: Theory
a: Pattern
well_formed a → Γ ⊢i ! ! a ---> a using AnyReasoning
  intros wfa.
Σ: Signature
Γ: Theory
a: Pattern
wfa: well_formed a
Γ ⊢i ! ! a ---> a using AnyReasoning
  toMLGoal.
Σ: Signature
Γ: Theory
a: Pattern
wfa: well_formed a
well_formed (! ! a ---> a)
Σ: Signature
Γ: Theory
a: Pattern
wfa: well_formed a
Γ Ⱶ
 ! ! a ---> a

  {
Σ: Signature
Γ: Theory
a: Pattern
wfa: well_formed a
well_formed (! ! a ---> a) wf_auto2. }
Σ: Signature
Γ: Theory
a: Pattern
wfa: well_formed a
Γ Ⱶ
 ! ! a ---> a

  mlTauto.
Defined.

#[local]
Lemma Peirce_taut {Σ : Signature} Γ a b:
  well_formed a ->
  well_formed b ->
  Γ ⊢i ((((a ---> b) ---> a) ---> a))
  using AnyReasoning.
Σ: Signature
Γ: Theory
a, b: Pattern
well_formed a
→ well_formed b
  → Γ ⊢i ((a ---> b) ---> a) ---> a using AnyReasoning
Proof.
Σ: Signature
Γ: Theory
a, b: Pattern
well_formed a
→ well_formed b
  → Γ ⊢i ((a ---> b) ---> a) ---> a using AnyReasoning
  intros wfa wfb.
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ ⊢i ((a ---> b) ---> a) ---> a using AnyReasoning
  toMLGoal.
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
well_formed (((a ---> b) ---> a) ---> a)
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ Ⱶ
 ((a ---> b) ---> a) ---> a

  {
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
well_formed (((a ---> b) ---> a) ---> a) wf_auto2. }
Σ: Signature
Γ: Theory
a, b: Pattern
wfa: well_formed a
wfb: well_formed b
Γ Ⱶ
 ((a ---> b) ---> a) ---> a

  mlTauto.
Defined.



Close Scope ml_scope.
Close Scope list_scope.
Close Scope string_scope.